Question

Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of f, of, g, of, minus, 1, .f(g(−1)).
f, of, x, equals, x, squared, plus, 2, x, minus, 5
f(x)=
x
2
+2x−5
g, of, x, equals, 5, x, minus, 4
g(x)=
5x−4

Answer 1

To solve for \( f(g(-1)) \) and then subsequently evaluate \( f(f(g(-1))) \), we will follow these steps:

1. Calculate \( g(-1) \).
2. Use the result from step 1 to calculate \( f(g(-1)) \).
3. Finally, use the result from step 2 to calculate \( f(f(g(-1))) \).

### Step 1: Calculate \( g(-1) \)

The function \( g(x) \) is defined as:

\[
g(x) = 5x - 4
\]

Substituting \( x = -1 \):

\[
g(-1) = 5(-1) - 4 = -5 - 4 = -9
\]

### Step 2: Calculate \( f(g(-1)) = f(-9) \)

Next, we will substitute the value of \( g(-1) \) into the function \( f(x) \). The function \( f(x) \) is defined as:

\[
f(x) = x^2 + 2x - 5
\]

Now substituting \( x = -9 \):

\[
f(-9) = (-9)^2 + 2(-9) - 5
\]
\[
= 81 - 18 - 5
\]
\[
= 81 - 18 - 5 = 58
\]

### Step 3: Calculate \( f(f(g(-1))) = f(58) \)

Now we will use the result from the previous step to find \( f(58) \):

\[
f(58) = 58^2 + 2(58) - 5
\]
\[
= 3364 + 116 - 5
\]
\[
= 3364 + 116 - 5 = 3475
\]

Thus, the value of \( f(f(g(-1))) \) is:

\[
\boxed{3475}
\]