The function f is defined for all real numbers by f(x) = \begin{cases} 3x - 1 & \text{if }… — GMAT

The function

f

is defined for all real numbers by

f(x) = \begin{cases} 3x - 1 & \text{if } x \le 4 \\ x^2 - k & \text{if } x > 4 \end{cases}

where

k

is a constant. If

f(f(3)) = 14

, what is the value of

k

Answer: 50

To solve for

k

, evaluate the composite function

f(f(3))

step-by-step. First, find

f(3)

. Since

3 \le 4

, we use the first branch of the definition:

f(3) = 3(3) - 1 = 8

. Next, evaluate

f(8)

. Since

8 > 4

, we use the second branch of the definition:

f(8) = 8^2 - k = 64 - k

. Given that

f(f(3)) = 14

, we set

64 - k = 14

, which yields

k = 50

Evaluate the inner function value f(3)

f(3) = 8

Because 3 is less than or equal to 4, the definition f(x) = 3x - 1 applies, yielding f(3) = 3(3) - 1 = 8.

Express f(f(3)) in terms of k

f(8) = 64 - k

Since the inner value is 8 and 8 > 4, the piecewise condition specifies using f(x) = x^2 - k, giving f(8) = 8^2 - k = 64 - k.

Solve the linear equation for k

k = 50

Setting the calculated expression equal to the given value 14 gives 64 - k = 14, which simplifies to k = 50.

Piecewise Function Evaluation and Parameter Resolution

Estimated Time:1m 30s

Question