Let ​f(x)equalsStartFraction 8 x minus 9 Over 6 EndFraction

Question

and ​g(x)equalsStartFraction 6 x plus 9 Over 8 EndFraction
.
​(a) Find left parenthesis f circle g right parenthesis left parenthesis x right parenthesis.
​(b) Find left parenthesis g circle f right parenthesis left parenthesis x right parenthesis.
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Part 1
​(a) left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find the compositions of the functions \( f(x) \) and \( g(x) \), we will first recall the definitions:

\[ f(x) = \frac{8x - 9}{6} \] \[ g(x) = \frac{6x + 9}{8} \]

(a) Find \( (f \circ g)(x) \)

This means we need to find \( f(g(x)) \).

1. Calculate \( g(x) \):

\[ g(x) = \frac{6x + 9}{8} \]

2. Substitute \( g(x) \) into \( f(x) \):

Now, substitute \( g(x) \) into \( f(x) \):

\[ f(g(x)) = f\left(\frac{6x + 9}{8}\right) = \frac{8\left(\frac{6x + 9}{8}\right) - 9}{6} \]

3. Simplify the expression:

\[ = \frac{(6x + 9) - 9}{6} = \frac{6x}{6} = x \]

Thus,

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

(b) Find \( (g \circ f)(x) \)

Now we need to find \( g(f(x)) \).

1. Calculate \( f(x) \):

We already have \( f(x) \):

\[ f(x) = \frac{8x - 9}{6} \]

2. Substitute \( f(x) \) into \( g(x) \):

Now substitute \( f(x) \) into \( g(x) \):

\[ g(f(x)) = g\left(\frac{8x - 9}{6}\right) = \frac{6\left(\frac{8x - 9}{6}\right) + 9}{8} \]

3. Simplify the expression:

\[ = \frac{8x - 9 + 9}{8} = \frac{8x}{8} = x \]

Thus,

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

Summary

\[ (f \circ g)(x) = x \quad \text{and} \quad (g \circ f)(x) = x \]

Both compositions return \( x \).