2580.22 – If . . . Then . . .

If f(x)=4xf(x) = 4^x, then f(x+1)−f(x)=⋯f(x+1) - f(x) = \cdots.

  1. 44

  2. f(x)f(x)

  3. 2f(x)2 f(x)

  4. 3f(x)3 f(x)

  5. 4f(x)4 f(x)

Solution

f(x+1)−f(x)=4x+1−4x=4⋅4x−4x=(4−1)4x=3⋅4x=3f(x)\begin{aligned}f(x+1) - f(x) &= 4^{x+1} - 4^x \\&= 4 \cdot 4^x - 4^x \\&= (4 - 1) 4^x \\&= 3 \cdot 4^x \\&= 3 f(x)\end{aligned}