Question

Use ​f(x)equals4 x squared and ​g(x)equalsStartFraction x minus 6 Over 2 EndFraction
to evaluate the expression.
a. left parenthesis f circle g right parenthesis left parenthesis 4 right parenthesis b. left parenthesis g circle f right parenthesis left parenthesis negative 5 right parenthesis

Answer 1

To evaluate the expressions \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we first need to define the functions \( f(x) \) and \( g(x) \):

- \( f(x) = 4x^2 \)
- \( g(x) = \frac{x - 6}{2} \)

### a. Evaluate \( (f \circ g)(4) \)

This means we first find \( g(4) \), and then plug that result into \( f(x) \).

1. **Calculate \( g(4) \)**:
\[
g(4) = \frac{4 - 6}{2} = \frac{-2}{2} = -1
\]

2. **Now calculate \( f(g(4)) = f(-1) \)**:
\[
f(-1) = 4(-1)^2 = 4(1) = 4
\]

Thus, \( (f \circ g)(4) = 4 \).

### b. Evaluate \( (g \circ f)(-5) \)

This means we first find \( f(-5) \), and then plug that result into \( g(x) \).

1. **Calculate \( f(-5) \)**:
\[
f(-5) = 4(-5)^2 = 4(25) = 100
\]

2. **Now calculate \( g(f(-5)) = g(100) \)**:
\[
g(100) = \frac{100 - 6}{2} = \frac{94}{2} = 47
\]

Thus, \( (g \circ f)(-5) = 47 \).

### Summary of Results
- \( (f \circ g)(4) = 4 \)
- \( (g \circ f)(-5) = 47 \)