Operations & Word Problems

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

Plug special inputs into the identity

A function f(x)f(x) satisfies f(x)f(y)=f(x+y)+f(x−y)f(x)f(y)=f(x+y)+f(x-y) for all numbers xx and yy, and f(1)=1f(1)=1. Find the value of 2f(0)+f(2)2f(0)+f(2).

FunctionsExpressions
Your answer
How to solve
Strategy Solve an Easier Related Problem — Rather than the general identity, solve easier special cases: y=0 makes x+y and x-y equal, and x=y makes x-y=0. With x=1 these give f(0) first and then f(2).
1STEP 1

Use x = 1, y = 0

Both x+y and x-y become 1.

f(1)f(0)=f(1)+f(1)→ 1 × f(0)=2 → f(0)=2
2STEP 2

Use x = 1, y = 1

Now x+y=2 and x-y=0, and f(0) is already known.

f(1)f(1)=f(2)+f(0)→ 1=f(2)+2 → f(2)=-1
3STEP 3

Combine

Put the two values into the expression.

2f(0)+f(2)=2×2+(-1)=3
Answer
3
Test another pair for consistency: x=2, y=0 gives f(2)f(0)=2f(2), that is (-1)×2=-2=2×(-1). It holds.
Takeaway

When a rule holds for all inputs, feed it the friendliest ones, like 0 and 1, first.

  • Use x = 1, y = 0
  • Use x = 1, y = 1
  • Combine
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