Question

Given f(x)=3x²-5 and g(x)=9-2x, find f(g(5)).

• A. -1
• B. 4
• C. 131
• D. -2
• E. 70

Hint:

Always follow the "inside-out" rule for composite functions. Solving for \(g(5)\) first simplifies the problem into two distinct, easier steps. Avoid the temptation to first find the general expression for \(f(g(x))\) and then substitute \(x=5\), as it can be more time-consuming and prone to algebraic errors.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem asks for the value of a composite function, \(f(g(x))\), at a specific point, \(x=5\). The process is to first evaluate the inner function, \(g(5)\), and then use the result as the input for the outer function, \(f(x)\).
Step 2: Detailed Explanation:
We need to find \(f(g(5))\).
First, let's find the value of the inner function, \(g(5)\).
The function \(g(x)\) is given by \(g(x) = 9 - 2x\).
Substitute \(x = 5\) into \(g(x)\):
\[ g(5) = 9 - 2(5) \] \[ g(5) = 9 - 10 \] \[ g(5) = -1 \] Now, we take this result, -1, and use it as the input for the function \(f(x)\).
We need to find \(f(g(5))\), which is now \(f(-1)\).
The function \(f(x)\) is given by \(f(x) = 3x^2 - 5\).
Substitute \(x = -1\) into \(f(x)\):
\[ f(-1) = 3(-1)^2 - 5 \] Remember that \((-1)^2 = 1\).
\[ f(-1) = 3(1) - 5 \] \[ f(-1) = 3 - 5 \] \[ f(-1) = -2 \] Step 3: Final Answer:
Therefore, the value of \(f(g(5))\) is -2.