Find ​(fcircle​g)(x) and ​(gcircle​f)(x).

Question

​f(x)equals5xminus5​, ​ g(x)equals6minus2x
Question content area bottom
Part 1
​(fcircle​g)(x)equals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \).

We have:

• \( f(x) = 5x - 5 \)
• \( g(x) = 6 - 2x \)

Now we'll find \( (f \circ g)(x) \):

\[ (f \circ g)(x) = f(g(x)) = f(6 - 2x) \]

Next, substitute \( 6 - 2x \) into \( f(x) \):

\[ f(6 - 2x) = 5(6 - 2x) - 5 \]

Now distribute \( 5 \):

\[ = 30 - 10x - 5 \]

Combine like terms:

\[ = 25 - 10x \]

So,

\[ (f \circ g)(x) = 25 - 10x \]

---

Now for \( (g \circ f)(x) \), we substitute \( f(x) \) into \( g(x) \):

\[ (g \circ f)(x) = g(f(x)) = g(5x - 5) \]

Substituting into \( g(x) \):

\[ g(5x - 5) = 6 - 2(5x - 5) \]

Distributing \( -2 \):

\[ = 6 - 10x + 10 \]

Combining like terms:

\[ = 16 - 10x \]

So,

\[ (g \circ f)(x) = 16 - 10x \]

In summary:

\[ (f \circ g)(x) = 25 - 10x \] \[ (g \circ f)(x) = 16 - 10x \]