Confirm that f and g are inverses by showing that

​f(g(x))equals=x

and

​g(f(x))equals=x.

​f(x)equals=StartFraction x minus 4 Over x EndFractionx−4x​,

​g(x)equals=negative StartFraction 4 Over x minus 1 EndFraction−4x−1

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Part 1

Is​ f(g(x)) equal to​ x?

Yes

No

Part 2

Is​ g(f(x)) equal to​ x?

No

Yes

To confirm that two functions \( f(x) \) and \( g(x) \) are inverses of each other, we need to show that both of the following are true:

1. \( f(g(x)) = x \)
2. \( g(f(x)) = x \)

We will go step by step to verify both conditions.

### Given Functions:
- \( f(x) = \frac{x - 4}{x} \)
- \( g(x) = -\frac{4}{x - 1} \)

### Part 1: Is \( f(g(x)) = x \)?

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

\[
f(g(x)) = f\left(-\frac{4}{x - 1}\right) = \frac{\left(-\frac{4}{x - 1}\right) - 4}{\left(-\frac{4}{x - 1}\right)}
\]

Simplify the numerator:

\[
\left(-\frac{4}{x - 1}\right) - 4 = \frac{-4 - 4(x - 1)}{x - 1} = \frac{-4 - 4x + 4}{x - 1} = \frac{-4x}{x - 1}
\]

Now, substitute this into the expression for \( f(g(x)) \):

\[
f(g(x)) = \frac{\frac{-4x}{x - 1}}{-\frac{4}{x - 1}} = x
\]

So, \( f(g(x)) = x \). The answer for Part 1 is **Yes**.

### Part 2: Is \( g(f(x)) = x \)?

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

\[
g(f(x)) = g\left(\frac{x - 4}{x}\right) = -\frac{4}{\frac{x - 4}{x} - 1}
\]

Simplify the denominator:

\[
\frac{x - 4}{x} - 1 = \frac{x - 4 - x}{x} = \frac{-4}{x}
\]

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

\[
g(f(x)) = -\frac{4}{\frac{-4}{x}} = x
\]

So, \( g(f(x)) = x \). The answer for Part 2 is **Yes**.

### Final Answer:
- Part 1: **Yes**
- Part 2: **Yes**

Both functions are indeed inverses of each other.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What are the general properties of inverse functions?
2. How can we algebraically find the inverse of a given function?
3. What is the geometric relationship between a function and its inverse on a graph?
4. How can we verify that a function is one-to-one before finding its inverse?
5. How does the domain and range of a function relate to the domain and range of its inverse?

**Tip**: Always check that the composition of a function with its inverse gives back the identity function, \( x \), to verify they are truly inverses.

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = (x - 4)/x, g(x) = -4/(x - 1). Part 1: Is f(g(x)) = x? Part 2: Is g(f(x)) = x?

Verify the inverse relationship between the functions f(x) = (x - 4)/x and g(x) = -4/(x - 1) by confirming that f(g(x)) = x and g(f(x)) = x. This algebra problem covers key concepts such as function composition and inverse functions, suitable for high school students in Grades 10-12.

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