Question

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Find ​(fcircle​g)(x) and ​(gcircle​f)(x).
​f(x)equals10xminus6​, ​ g(x)equals7minus2x
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Part 1
​(fcircle​g)(x)equals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find \((f \circ g)(x)\) which is the composition of functions \(f\) and \(g\), we will substitute \(g(x)\) into \(f(x)\).

We are given:

- \(f(x) = 10x - 6\)
- \(g(x) = 7 - 2x\)

Now we compute \((f \circ g)(x)\):

\[
(f \circ g)(x) = f(g(x)) = f(7 - 2x)
\]

Now, we substitute \(7 - 2x\) into \(f(x)\):

\[
f(7 - 2x) = 10(7 - 2x) - 6
\]

Now, simplify:

\[
= 10 \cdot 7 - 10 \cdot 2x - 6
\]
\[
= 70 - 20x - 6
\]
\[
= 64 - 20x
\]

So,
\[
(f \circ g)(x) = 64 - 20x
\]

---

Now, let's find \((g \circ f)(x)\):

\[
(g \circ f)(x) = g(f(x)) = g(10x - 6)
\]

Substituting \(10x - 6\) into \(g(x)\):

\[
g(10x - 6) = 7 - 2(10x - 6)
\]

Now, simplify:

\[
= 7 - 20x + 12
\]
\[
= 19 - 20x
\]

So,
\[
(g \circ f)(x) = 19 - 20x
\]

In summary, the results are:

1. **\((f \circ g)(x) = 64 - 20x\)**
2. **\((g \circ f)(x) = 19 - 20x\)**