Practice · Find the x that makes the input right · Sensim Math

Variant 1 easy

A function $f$ satisfies $f\left(\frac{x+6}{x-4}\right)=x^2+3x-4$ . Find the value of $f(-4)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x+6}{x-4}$ . To get $f(-4)$ , find the $x$ that makes that expression equal $-4$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x+6}{x-4}\right)=x^2+3x-4$ .

Unknowns

The value of $f(-4)$ .

Constraints

The input of $f$ is the whole fraction, so we need $x-4 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $-4$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input -4

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

-4

.

\frac{x+6}{x-4}=-4

f(-4)

means the output when the input is

-4

, so first make the input

-4

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

x-4

and collect terms.

x+6=-4x+16 \Rightarrow 5x=10 \Rightarrow x=2

Check:

\frac{2+6}{2-4}=\frac{8}{-2}=-4

, and the denominator

-2

is not zero.

3Evaluate the right side at x = 2

#13 Convert to Algebra 8.EE.C.7

Substitute

x=2

into

x^2+3x-4

.

f(-4)=2^2+3\times2-4=4+6-4=6

When

x=2

the left side is

f(-4)

, so the right side is its value.

Answer:

6

4 · Reviewdoes it hold up?

Substituting $x=-4$ directly would be the trap: it gives $f\left(\frac{2}{-8}\right)=f(-\frac{1}{4})$ , not $f(-4)$ . Our $x=2$ really makes the input $-4$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $-1\frac{1}{2}$ , $x=1$ gives $-2\frac{1}{3}$ , $x=2$ gives $-4$ . The guess $x=2$ hits, and the right side is $4+6-4=6$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(-4)$ as the output for the input $-4$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x+6=-4x+16$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 2 easy

A function $f$ satisfies $f\left(\frac{x+3}{3x-1}\right)=-x^2-2x-4$ . Find the value of $f(1)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x+3}{3x-1}$ . To get $f(1)$ , find the $x$ that makes that expression equal $1$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x+3}{3x-1}\right)=-x^2-2x-4$ .

Unknowns

The value of $f(1)$ .

Constraints

The input of $f$ is the whole fraction, so we need $3x-1 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $1$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input 1

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

1

.

\frac{x+3}{3x-1}=1

f(1)

means the output when the input is

1

, so first make the input

1

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

3x-1

and collect terms.

x+3=3x-1 \Rightarrow -2x=-4 \Rightarrow x=2

Check:

\frac{2+3}{3\times2-1}=\frac{5}{5}=1

, and the denominator

5

is not zero.

3Evaluate the right side at x = 2

#13 Convert to Algebra 8.EE.C.7

Substitute

x=2

into

-x^2-2x-4

. Note that

-x^2

means

-(x^2)

.

f(1)=-2^2-2\times2-4=-4-4-4=-12

When

x=2

the left side is

f(1)

, so the right side is its value.

Answer:

-12

4 · Reviewdoes it hold up?

Substituting $x=1$ directly would be the trap: it gives $f\left(\frac{4}{2}\right)=f(2)$ , not $f(1)$ . Our $x=2$ really makes the input $1$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $-3$ , $x=1$ gives $2$ , $x=2$ gives $1$ . The guess $x=2$ hits, and the right side is $-4-4-4=-12$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(1)$ as the output for the input $1$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x+3=3x-1$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 3 easy

A function $f$ satisfies $f\left(\frac{x-8}{2x-1}\right)=-x^2+x-4$ . Find the value of $f(-2)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x-8}{2x-1}$ . To get $f(-2)$ , find the $x$ that makes that expression equal $-2$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x-8}{2x-1}\right)=-x^2+x-4$ .

Unknowns

The value of $f(-2)$ .

Constraints

The input of $f$ is the whole fraction, so we need $2x-1 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $-2$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input -2

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

-2

.

\frac{x-8}{2x-1}=-2

f(-2)

means the output when the input is

-2

, so first make the input

-2

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

2x-1

and collect terms.

x-8=-4x+2 \Rightarrow 5x=10 \Rightarrow x=2

Check:

\frac{2-8}{2\times2-1}=\frac{-6}{3}=-2

, and the denominator

3

is not zero.

3Evaluate the right side at x = 2

#13 Convert to Algebra 8.EE.C.7

Substitute

x=2

into

-x^2+x-4

. Note that

-x^2

means

-(x^2)

.

f(-2)=-2^2+2-4=-4+2-4=-6

When

x=2

the left side is

f(-2)

, so the right side is its value.

Answer:

-6

4 · Reviewdoes it hold up?

Substituting $x=-2$ directly would be the trap: it gives $f\left(\frac{-10}{-5}\right)=f(2)$ , not $f(-2)$ . Our $x=2$ really makes the input $-2$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $8$ , $x=1$ gives $-7$ , $x=2$ gives $-2$ . The guess $x=2$ hits, and the right side is $-4+2-4=-6$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(-2)$ as the output for the input $-2$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x-8=-4x+2$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 4 easy

A function $f$ satisfies $f\left(\frac{x-9}{x-5}\right)=x^2-2x-4$ . Find the value of $f(2)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x-9}{x-5}$ . To get $f(2)$ , find the $x$ that makes that expression equal $2$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x-9}{x-5}\right)=x^2-2x-4$ .

Unknowns

The value of $f(2)$ .

Constraints

The input of $f$ is the whole fraction, so we need $x-5 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $2$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input 2

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

2

.

\frac{x-9}{x-5}=2

f(2)

means the output when the input is

2

, so first make the input

2

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

x-5

and collect terms.

x-9=2x-10 \Rightarrow -x=-1 \Rightarrow x=1

Check:

\frac{1-9}{1-5}=\frac{-8}{-4}=2

, and the denominator

-4

is not zero.

3Evaluate the right side at x = 1

#13 Convert to Algebra 8.EE.C.7

Substitute

x=1

into

x^2-2x-4

.

f(2)=1^2-2\times1-4=1-2-4=-5

When

x=1

the left side is

f(2)

, so the right side is its value.

Answer:

-5

4 · Reviewdoes it hold up?

Substituting $x=2$ directly would be the trap: it gives $f\left(\frac{-7}{-3}\right)=f(2\frac{1}{3})$ , not $f(2)$ . Our $x=1$ really makes the input $2$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $1\frac{4}{5}$ , $x=1$ gives $2$ . The guess $x=1$ hits, and the right side is $1-2-4=-5$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(2)$ as the output for the input $2$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x-9=2x-10$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 5 medium

A function $f$ satisfies $f\left(\frac{x-5}{2x-1}\right)=x^2-2x-4$ . Find the value of $f(-1)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x-5}{2x-1}$ . To get $f(-1)$ , find the $x$ that makes that expression equal $-1$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x-5}{2x-1}\right)=x^2-2x-4$ .

Unknowns

The value of $f(-1)$ .

Constraints

The input of $f$ is the whole fraction, so we need $2x-1 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $-1$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input -1

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

-1

.

\frac{x-5}{2x-1}=-1

f(-1)

means the output when the input is

-1

, so first make the input

-1

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

2x-1

and collect terms.

x-5=-2x+1 \Rightarrow 3x=6 \Rightarrow x=2

Check:

\frac{2-5}{2\times2-1}=\frac{-3}{3}=-1

, and the denominator

3

is not zero.

3Evaluate the right side at x = 2

#13 Convert to Algebra 8.EE.C.7

Substitute

x=2

into

x^2-2x-4

.

f(-1)=2^2-2\times2-4=4-4-4=-4

When

x=2

the left side is

f(-1)

, so the right side is its value.

Answer:

-4

4 · Reviewdoes it hold up?

Substituting $x=-1$ directly would be the trap: it gives $f\left(\frac{-6}{-3}\right)=f(2)$ , not $f(-1)$ . Our $x=2$ really makes the input $-1$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $5$ , $x=1$ gives $-4$ , $x=2$ gives $-1$ . The guess $x=2$ hits, and the right side is $4-4-4=-4$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(-1)$ as the output for the input $-1$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x-5=-2x+1$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 6 medium

A function $f$ satisfies $f\left(\frac{x+2}{3x-4}\right)=x^2+3x+5$ . Find the value of $f(1)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x+2}{3x-4}$ . To get $f(1)$ , find the $x$ that makes that expression equal $1$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x+2}{3x-4}\right)=x^2+3x+5$ .

Unknowns

The value of $f(1)$ .

Constraints

The input of $f$ is the whole fraction, so we need $3x-4 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $1$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input 1

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

1

.

\frac{x+2}{3x-4}=1

f(1)

means the output when the input is

1

, so first make the input

1

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

3x-4

and collect terms.

x+2=3x-4 \Rightarrow -2x=-6 \Rightarrow x=3

Check:

\frac{3+2}{3\times3-4}=\frac{5}{5}=1

, and the denominator

5

is not zero.

3Evaluate the right side at x = 3

#13 Convert to Algebra 8.EE.C.7

Substitute

x=3

into

x^2+3x+5

.

f(1)=3^2+3\times3+5=9+9+5=23

When

x=3

the left side is

f(1)

, so the right side is its value.

Answer:

23

4 · Reviewdoes it hold up?

Substituting $x=1$ directly would be the trap: it gives $f\left(\frac{3}{-1}\right)=f(-3)$ , not $f(1)$ . Our $x=3$ really makes the input $1$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $-\frac{1}{2}$ , $x=1$ gives $-3$ , $x=2$ gives $2$ , $x=3$ gives $1$ . The guess $x=3$ hits, and the right side is $9+9+5=23$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(1)$ as the output for the input $1$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x+2=3x-4$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 7 medium

A function $f$ satisfies $f\left(\frac{x+5}{3x-5}\right)=-x^2-2x-4$ . Find the value of $f(2)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x+5}{3x-5}$ . To get $f(2)$ , find the $x$ that makes that expression equal $2$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x+5}{3x-5}\right)=-x^2-2x-4$ .

Unknowns

The value of $f(2)$ .

Constraints

The input of $f$ is the whole fraction, so we need $3x-5 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $2$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input 2

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

2

.

\frac{x+5}{3x-5}=2

f(2)

means the output when the input is

2

, so first make the input

2

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

3x-5

and collect terms.

x+5=6x-10 \Rightarrow -5x=-15 \Rightarrow x=3

Check:

\frac{3+5}{3\times3-5}=\frac{8}{4}=2

, and the denominator

4

is not zero.

3Evaluate the right side at x = 3

#13 Convert to Algebra 8.EE.C.7

Substitute

x=3

into

-x^2-2x-4

. Note that

-x^2

means

-(x^2)

.

f(2)=-3^2-2\times3-4=-9-6-4=-19

When

x=3

the left side is

f(2)

, so the right side is its value.

Answer:

-19

4 · Reviewdoes it hold up?

Substituting $x=2$ directly would be the trap: it gives $f\left(\frac{7}{1}\right)=f(7)$ , not $f(2)$ . Our $x=3$ really makes the input $2$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $-1$ , $x=1$ gives $-3$ , $x=2$ gives $7$ , $x=3$ gives $2$ . The guess $x=3$ hits, and the right side is $-9-6-4=-19$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(2)$ as the output for the input $2$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x+5=6x-10$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 8 medium

A function $f$ satisfies $f\left(\frac{x-10}{x+2}\right)=2x^2+x+3$ . Find the value of $f(-1)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x-10}{x+2}$ . To get $f(-1)$ , find the $x$ that makes that expression equal $-1$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x-10}{x+2}\right)=2x^2+x+3$ .

Unknowns

The value of $f(-1)$ .

Constraints

The input of $f$ is the whole fraction, so we need $x+2 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $-1$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input -1

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

-1

.

\frac{x-10}{x+2}=-1

f(-1)

means the output when the input is

-1

, so first make the input

-1

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

x+2

and collect terms.

x-10=-x-2 \Rightarrow 2x=8 \Rightarrow x=4

Check:

\frac{4-10}{4+2}=\frac{-6}{6}=-1

, and the denominator

6

is not zero.

3Evaluate the right side at x = 4

#13 Convert to Algebra 8.EE.C.7

Substitute

x=4

into

2x^2+x+3

.

f(-1)=2\times4^2+4+3=32+4+3=39

When

x=4

the left side is

f(-1)

, so the right side is its value.

Answer:

39

4 · Reviewdoes it hold up?

Substituting $x=-1$ directly would be the trap: it gives $f\left(\frac{-11}{1}\right)=f(-11)$ , not $f(-1)$ . Our $x=4$ really makes the input $-1$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $-5$ , $x=1$ gives $-3$ , $x=2$ gives $-2$ , $x=3$ gives $-1\frac{2}{5}$ , $x=4$ gives $-1$ . The guess $x=4$ hits, and the right side is $32+4+3=39$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(-1)$ as the output for the input $-1$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x-10=-x-2$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 9 hard

A function $f$ satisfies $f\left(\frac{x+6}{2x+4}\right)=2x^2+x+3$ . Find the value of $f(1)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x+6}{2x+4}$ . To get $f(1)$ , find the $x$ that makes that expression equal $1$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x+6}{2x+4}\right)=2x^2+x+3$ .

Unknowns

The value of $f(1)$ .

Constraints

The input of $f$ is the whole fraction, so we need $2x+4 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $1$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input 1

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

1

.

\frac{x+6}{2x+4}=1

f(1)

means the output when the input is

1

, so first make the input

1

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

2x+4

and collect terms.

x+6=2x+4 \Rightarrow -x=-2 \Rightarrow x=2

Check:

\frac{2+6}{2\times2+4}=\frac{8}{8}=1

, and the denominator

8

is not zero.

3Evaluate the right side at x = 2

#13 Convert to Algebra 8.EE.C.7

Substitute

x=2

into

2x^2+x+3

.

f(1)=2\times2^2+2+3=8+2+3=13

When

x=2

the left side is

f(1)

, so the right side is its value.

Answer:

13

4 · Reviewdoes it hold up?

Substituting $x=1$ directly would be the trap: it gives $f\left(\frac{7}{6}\right)=f(1\frac{1}{6})$ , not $f(1)$ . Our $x=2$ really makes the input $1$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $1\frac{1}{2}$ , $x=1$ gives $1\frac{1}{6}$ , $x=2$ gives $1$ . The guess $x=2$ hits, and the right side is $8+2+3=13$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(1)$ as the output for the input $1$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x+6=2x+4$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 10 hard

A function $f$ satisfies $f\left(\frac{x-8}{3x-4}\right)=-x^2+3x-4$ . Find the value of $f(-3)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x-8}{3x-4}$ . To get $f(-3)$ , find the $x$ that makes that expression equal $-3$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x-8}{3x-4}\right)=-x^2+3x-4$ .

Unknowns

The value of $f(-3)$ .

Constraints

The input of $f$ is the whole fraction, so we need $3x-4 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $-3$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input -3

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

-3

.

\frac{x-8}{3x-4}=-3

f(-3)

means the output when the input is

-3

, so first make the input

-3

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

3x-4

and collect terms.

x-8=-9x+12 \Rightarrow 10x=20 \Rightarrow x=2

Check:

\frac{2-8}{3\times2-4}=\frac{-6}{2}=-3

, and the denominator

2

is not zero.

3Evaluate the right side at x = 2

#13 Convert to Algebra 8.EE.C.7

Substitute

x=2

into

-x^2+3x-4

. Note that

-x^2

means

-(x^2)

.

f(-3)=-2^2+3\times2-4=-4+6-4=-2

When

x=2

the left side is

f(-3)

, so the right side is its value.

Answer:

-2

4 · Reviewdoes it hold up?

Substituting $x=-3$ directly would be the trap: it gives $f\left(\frac{-11}{-13}\right)=f(\frac{11}{13})$ , not $f(-3)$ . Our $x=2$ really makes the input $-3$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $2$ , $x=1$ gives $7$ , $x=2$ gives $-3$ . The guess $x=2$ hits, and the right side is $-4+6-4=-2$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(-3)$ as the output for the input $-3$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x-8=-9x+12$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 11 hard

A function $f$ satisfies $f\left(\frac{x-1}{3x-5}\right)=2x^2+3x+3$ . Find the value of $f(1)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x-1}{3x-5}$ . To get $f(1)$ , find the $x$ that makes that expression equal $1$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x-1}{3x-5}\right)=2x^2+3x+3$ .

Unknowns

The value of $f(1)$ .

Constraints

The input of $f$ is the whole fraction, so we need $3x-5 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $1$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input 1

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

1

.

\frac{x-1}{3x-5}=1

f(1)

means the output when the input is

1

, so first make the input

1

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

3x-5

and collect terms.

x-1=3x-5 \Rightarrow -2x=-4 \Rightarrow x=2

Check:

\frac{2-1}{3\times2-5}=\frac{1}{1}=1

, and the denominator

1

is not zero.

3Evaluate the right side at x = 2

#13 Convert to Algebra 8.EE.C.7

Substitute

x=2

into

2x^2+3x+3

.

f(1)=2\times2^2+3\times2+3=8+6+3=17

When

x=2

the left side is

f(1)

, so the right side is its value.

Answer:

17

4 · Reviewdoes it hold up?

Substituting $x=1$ directly would be the trap: it gives $f\left(\frac{0}{-2}\right)=f(0)$ , not $f(1)$ . Our $x=2$ really makes the input $1$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $\frac{1}{5}$ , $x=1$ gives $0$ , $x=2$ gives $1$ . The guess $x=2$ hits, and the right side is $8+6+3=17$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(1)$ as the output for the input $1$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x-1=3x-5$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Variant 12 hard

A function $f$ satisfies $f\left(\frac{x-10}{2x-5}\right)=x^2+3x+5$ . Find the value of $f(-2)$ .

Your answer

Show solution

1 · Understandwhat's really being asked

The rule does not give $f$ of a plain $x$ ; it gives $f$ of the expression $\frac{x-10}{2x-5}$ . To get $f(-2)$ , find the $x$ that makes that expression equal $-2$ , then evaluate the right-hand side at that $x$ .

Givens

$f\left(\frac{x-10}{2x-5}\right)=x^2+3x+5$ .

Unknowns

The value of $f(-2)$ .

Constraints

The input of $f$ is the whole fraction, so we need $2x-5 \neq 0$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #13 Convert to Algebra

Work backwards from the input we want: set the inside expression equal to $-2$ , solve that linear equation for $x$ , and only then substitute into the right-hand side.

3 · Execute3 carry out the plan

1Ask which x gives input -2

#11 Work Backwards 8.F.A.1

We need the fraction inside

f

to equal

-2

.

\frac{x-10}{2x-5}=-2

f(-2)

means the output when the input is

-2

, so first make the input

-2

.

2Solve for x

#13 Convert to Algebra 8.EE.C.7

Multiply both sides by

2x-5

and collect terms.

x-10=-4x+10 \Rightarrow 5x=20 \Rightarrow x=4

Check:

\frac{4-10}{2\times4-5}=\frac{-6}{3}=-2

, and the denominator

3

is not zero.

3Evaluate the right side at x = 4

#13 Convert to Algebra 8.EE.C.7

Substitute

x=4

into

x^2+3x+5

.

f(-2)=4^2+3\times4+5=16+12+5=33

When

x=4

the left side is

f(-2)

, so the right side is its value.

Answer:

33

4 · Reviewdoes it hold up?

Substituting $x=-2$ directly would be the trap: it gives $f\left(\frac{-12}{-9}\right)=f(1\frac{1}{3})$ , not $f(-2)$ . Our $x=4$ really makes the input $-2$ .

Another way: Guess and check small values of $x$ : $x=0$ gives $2$ , $x=1$ gives $3$ , $x=2$ gives $8$ , $x=3$ gives $-7$ , $x=4$ gives $-2$ . The guess $x=4$ hits, and the right side is $16+12+5=33$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Understanding $f(-2)$ as the output for the input $-2$.
8.EE.C.7 Solve linear equations in one variable — Solving the linear equation $x-10=-4x+10$ and evaluating the expression.

💡Takeaway. When the rule is written for a disguised input, solve for the x that makes the input what you want.

Find the x that makes the input right · 12 practice problems

Generated variants — 12

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