Question

Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of f, of, g, of, minus, 2, .f(g(−2)).
f, of, x, equals, 3, x, squared, minus, 7, x, plus, 15
f(x)=
3x
2
−7x+15
g, of, x, equals, x, plus, 7
g(x)=
x+7

Answer 1

To solve the expression \( f(g(-2)) \) step-by-step, we first need to calculate \( g(-2) \) using the definition of \( g(x) \).

1. **Calculate \( g(-2) \)**:
\[
g(x) = x + 7
\]
Substituting \(-2\) into \(g(x)\):
\[
g(-2) = -2 + 7 = 5
\]

2. **Now, calculate \( f(g(-2)) = f(5) \)**:
Using the definition of \( f(x) \):
\[
f(x) = 3x^2 - 7x + 15
\]
Substituting \(5\) into \(f(x)\):
\[
f(5) = 3(5^2) - 7(5) + 15
\]
First, compute \(5^2\):
\[
5^2 = 25
\]
Now calculate \(f(5)\):
\[
f(5) = 3(25) - 35 + 15 = 75 - 35 + 15
\]
Combining these:
\[
75 - 35 = 40
\]
\[
40 + 15 = 55
\]
So, \(f(5) = 55\).

3. **Finally, we need to determine \(f(g(-2))\) and we have found it equals \(55\)**. The original question seems to ask for \(f(g(-2))\) without additional operations such as subtraction or further composition, so:
\[
\text{Value of } f(g(-2)) = 55
\]

In conclusion, the answer is:
\[
\boxed{55}
\]