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Question for-th

2.1 Nonlinear Functions - Exponential growth and decay functions
0:00
xf of x
1a
2a, to the fifth power.
3a, to the ninth power


For the exponential function f, the table above shows several values of x and their corresponding values of f of x, where a is a constant greater than 1. If k is a constant and f of k, equals a, to the twenty ninth power, what is the value of k ?

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For the exponential function f , the table above shows several values of x and their

Hard-difficulty · SAT Math · Nonlinear Functions — Exponential growth and decay functions. Read the question above, select your answer, and check the full explanation below to understand exactly why the correct choice works.

Answer explanation

The correct answer is 8. The values of f of x for the exponential function f shown in the table increase by a factor of a, to the fourth power for each increase of 1 in x. This relationship can be represented by the equation f of x equals, a, raised to the 4 x plus b power, where b is a constant. It’s given that when x equals 2, f of x equals, a, to the fifth power.  Substituting 2 for x and a, to the fifth power for f of x into f of x equals, a, raised to the 4 x plus b power yields a, to the fifth power equals, a, raised to the 4 times 2, plus b power. Since 4 times 2, plus b, equals 5, it follows that b equals negative 3. Thus, an equation that defines the function f is f of x equals, a, raised to the 4 x minus 3 power. It follows that the value of k such that f of k equals, a, to the twenty ninth power can be found by solving the equation 4 k minus 3, equals 29, which yields k equals 8.

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