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

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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. \((g \circ f)(x) = 19 - 20x\)