Depth · one concept from first principles · step 2 of 4

Find the x that makes the input right

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

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Your answer

How to solve

Strategy Work Backwards — 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.

1STEP 1

Ask which x gives input -2

We need the fraction inside f to equal -2.

(x-8)/(2x-1)=-2

2STEP 2

Solve for x

Multiply both sides by 2x-1 and collect terms.

x-8=-4x+2 → 5x=10 → x=2

3STEP 3

Evaluate the right side at x = 2

Substitute x=2 into -x²+x-4. Note that -x² means -(x²).

f(-2)=-2²+2-4=-4+2-4=-6

Answer

-6

Substituting x=-2 directly would be the trap: it gives f(-10/-5)=f(2), not f(-2). Our x=2 really makes the input -2.

Takeaway

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

Ask which x gives input -2
Solve for x
Evaluate the right side at x = 2

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