The function f is defined by f(x) = a^x + b, where a and b... | SAT Nonlinear functions Question, Answer Explained

Math · Advanced MathHard

The function $f$ is defined by $f(x) = a^x + b$ , where $a$ and $b$ are positive constants. The graph of $y = f(x)$ in the $xy$ -plane passes through the points $(0, 7)$ and $(2, 22)$ . What is the value of $f(3)$ ?

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Answer: 70

At $x = 0$ , $a^0 = 1$ , so $f(0) = 1 + b = 7$ and $b = 6$ . Then $f(2) = a^2 + 6 = 22$ , so $a^2 = 16$ and $a = 4$ , because $a$ is positive. So $f(3) = 4^3 + 6 = 64 + 6 = 70$ .

Assuming $f(0) = b$ forgets that $a^0 = 1$ , which leads to $b = 7$ instead of $6$ .

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