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Use the pair of functions to find \( f(g(x)) \) and \( g(f(x)) \). Simplify your answers.

[tex]\[
f(x) = \frac{1}{x-3}, \quad g(x) = \frac{6}{x} + 3
\][/tex]

Sagot :

GinnyAnswer
Given the functions:
[tex]\[ f(x) = \frac{1}{x - 3} \][/tex]
[tex]\[ g(x) = \frac{6}{x} + 3 \][/tex]

### Finding \( f(g(x)) \)

Substitute \( g(x) \) into \( f(x) \):
[tex]\[ f(g(x)) = f\left( \frac{6}{x} + 3 \right) \][/tex]

Now, use the definition of \( f(x) \):
[tex]\[ f(g(x)) = \frac{1}{\left( \frac{6}{x} + 3 \right) - 3} \][/tex]

Simplify inside the denominator:
[tex]\[ f(g(x)) = \frac{1}{\frac{6}{x} + 3 - 3} \][/tex]
[tex]\[ f(g(x)) = \frac{1}{\frac{6}{x}} \][/tex]
[tex]\[ f(g(x)) = \frac{x}{6} \][/tex]

So, \( f(g(x)) \) simplifies to:
[tex]\[ f(g(x)) = \frac{x}{6} \][/tex]

### Finding \( g(f(x)) \)

Substitute \( f(x) \) into \( g(x) \):
[tex]\[ g(f(x)) = g\left( \frac{1}{x - 3} \right) \][/tex]

Now, use the definition of \( g(x) \):
[tex]\[ g(f(x)) = \frac{6}{\frac{1}{x - 3}} + 3 \][/tex]

Simplify the expression inside \( g(x) \):
[tex]\[ g(f(x)) = 6(x - 3) + 3 \][/tex]

Distribute and combine like terms:
[tex]\[ g(f(x)) = 6x - 18 + 3 \][/tex]
[tex]\[ g(f(x)) = 6x - 15 \][/tex]

So, \( g(f(x)) \) simplifies to:
[tex]\[ g(f(x)) = 6x - 15 \][/tex]

### Final Results:

Thus, we have:
[tex]\[ f(g(x)) = \frac{x}{6} \][/tex]
[tex]\[ g(f(x)) = 6x - 15 \][/tex]
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