The function f is defined by f(x) = 2^x, and the function g... | SAT Nonlinear functions Question, Answer Explained

Math · Advanced MathHard

The function $f$ is defined by $f(x) = 2^x$ , and the function $g$ is defined by $g(x) = f(x + 3)$ . For all values of $x$ , $g(x) = k \cdot f(x)$ , where $k$ is a constant. What is the value of $k$ ?

$3$

$5$

$6$

$8$

Show answer and explanation

Answer: D

By the definition of $g$ , $g(x) = f(x + 3) = 2^{x + 3} = 2^x \cdot 2^3 = 8 \cdot 2^x$ . So $g(x) = 8f(x)$ and $k = 8$ .

Choice A uses the size of the shift, $3$ , but adding $3$ to the exponent multiplies the value by $2^3$ .

Why the other choices are wrong

A. 3 is the size of the shift. Adding 3 to the exponent multiplies the value by 2 × 2 × 2 = 8.
B. This adds 2 and 3. Adding 3 to the exponent multiplies the value by 2 three times.
C. This multiplies 2 by 3. Adding 3 to the exponent multiplies by 2 × 2 × 2, not 2 × 3.

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