For the exponential function g(x) = k \cdot a^x, where k and a are positive constants, the… — SAT

For the exponential function $g(x) = k \cdot a^x$ , where $k$ and $a$ are positive constants, the value of $g(x + 1)$ is $20\%$ greater than the value of $g(x - 1)$ for all real numbers $x$ . If $g(2) = 15$ , what is the value of $g(6)$ ?

A
21.0
21.6Answer
C
31.104
D
36.0

Answer

21.6

The correct value of 21.6 is obtained by setting up the ratio

g(x+1)/g(x-1) = a^2

. Since

g(x+1)

20\%

greater than

g(x-1)

, this ratio is equal to 1.20, so

a^2 = 1.20

. Since

g(6) = g(2) \cdot a^4 = 15 \cdot (a^2)^2

, substituting

a^2 = 1.20

gives

15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.6

Step-by-Step Solution

Express the values of the function at x + 1 and x - 1 using the definition of g(x).

g(x + 1) = k \cdot a^{x + 1}

and

g(x - 1) = k \cdot a^{x - 1}

To set up the mathematical relationship between the two values based on the function definition.

Apply the condition that g(x + 1) is 20% greater than g(x - 1).

k \cdot a^{x + 1} = 1.20 \cdot k \cdot a^{x - 1}

, which simplifies to

a^2 = 1.20

To find the factor of growth over an interval of 2 units of x.

Relate the value of g(6) to the given value of g(2).

g(6) = g(2) \cdot a^4 = g(2) \cdot (a^2)^2

To write the unknown value in terms of the known value and the determined factor

a^2

using exponent rules.

Substitute the known values into the equation to calculate the result.

g(6) = 15 \cdot (1.20)^2 = 15 \cdot 1.44 = 21.6

To calculate the final numerical value of g(6).

Key Concept

Exponential Functions and Equations

Estimated Time:2m 30s

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