Let
f
(
x
)
=
1
x
−
5
f
(
x
)
=
1
x
-
5
and
g
(
x
)
=
2
x
+
5
g
(
x
)
=
2
x
+
5
.

Find the following functions. Simplify your answers.

f
(
g
(
x
)
)
=
f
(
g
(
x
)
)
=
,

g
(
f
(
x
)
)
=
g
(
f
(
x
)
)
=
.

We are given two functions:

\[
f(x) = \frac{1}{x} - 5
\]
and
\[
g(x) = \frac{2}{x} + 5.
\]

We are asked to find two composite functions: \( f(g(x)) \) and \( g(f(x)) \).

### 1. Find \( f(g(x)) \):

To find \( f(g(x)) \), we substitute \( g(x) \) into the function \( f(x) \):

\[
f(g(x)) = f\left(\frac{2}{x} + 5\right).
\]

Now, substitute \( \frac{2}{x} + 5 \) into \( f(x) = \frac{1}{x} - 5 \):

\[
f(g(x)) = \frac{1}{\frac{2}{x} + 5} - 5.
\]

Simplify the first term:

\[
\frac{1}{\frac{2}{x} + 5} = \frac{x}{2 + 5x}.
\]

Thus,

\[
f(g(x)) = \frac{x}{2 + 5x} - 5.
\]

Now subtract \( 5 \) from the expression:

\[
f(g(x)) = \frac{x}{2 + 5x} - \frac{5(2 + 5x)}{2 + 5x} = \frac{x - 5(2 + 5x)}{2 + 5x}.
\]

Simplifying the numerator:

\[
x - 5(2 + 5x) = x - 10 - 25x = -24x - 10.
\]

Thus,

\[
f(g(x)) = \frac{-24x - 10}{2 + 5x}.
\]

### 2. Find \( g(f(x)) \):

Now, substitute \( f(x) = \frac{1}{x} - 5 \) into \( g(x) = \frac{2}{x} + 5 \):

\[
g(f(x)) = g\left(\frac{1}{x} - 5\right).
\]

Substitute into \( g(x) \):

\[
g(f(x)) = \frac{2}{\frac{1}{x} - 5} + 5.
\]

Simplify the first term:

\[
\frac{2}{\frac{1}{x} - 5} = \frac{2x}{1 - 5x}.
\]

Thus,

\[
g(f(x)) = \frac{2x}{1 - 5x} + 5.
\]

Now express \( 5 \) as \( \frac{5(1 - 5x)}{1 - 5x} \):

\[
g(f(x)) = \frac{2x}{1 - 5x} + \frac{5(1 - 5x)}{1 - 5x} = \frac{2x + 5(1 - 5x)}{1 - 5x}.
\]

Simplifying the numerator:

\[
2x + 5(1 - 5x) = 2x + 5 - 25x = -23x + 5.
\]

Thus,

\[
g(f(x)) = \frac{-23x + 5}{1 - 5x}.
\]

### Final Results:

\[
f(g(x)) = \frac{-24x - 10}{2 + 5x}
\]
and
\[
g(f(x)) = \frac{-23x + 5}{1 - 5x}.
\]

Would you like any further details or clarifications?

Here are some related questions you might consider:
1. What is the domain of \( f(g(x)) \)?
2. What is the domain of \( g(f(x)) \)?
3. Can you find the inverse of \( f(x) \)?
4. Can you find the inverse of \( g(x) \)?
5. What happens when you compose \( f(x) \) and \( g(x) \) in the opposite order?

**Tip**: When composing functions, always check the domains to ensure the resulting function is valid!

Let f(x) = 1/x - 5 and g(x) = 2/x + 5. Find the following functions. Simplify your answers. f(g(x)) = , g(f(x)) = .

Learn how to compose the functions f(x) = 1/x - 5 and g(x) = 2/x + 5. This step-by-step solution simplifies the composite functions f(g(x)) and g(f(x)), covering key algebraic techniques in rational function composition. Suitable for students in Grades 10-12.

Math Problem Statement

Let f ( x )

1 x − 5 f ( x )

1 x

5 and g ( x )

2 x + 5 g ( x )

f ( g ( x ) )

f ( g ( x ) )

g ( f ( x ) )

g ( f ( x ) )

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