Question

determine which pair of functions are inverses functions
a.f(x)=x-4
g(x)=x+4

b.f(x)=x-4
g(x)=x-4/4

c.f(x)=x-4
g(x)=4x-1

Answer 1

To determine whether two functions are inverse functions, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

Let's analyze each pair:

### a.
- \( f(x) = x - 4 \)
- \( g(x) = x + 4 \)

Calculating \( f(g(x)) \):
\[
f(g(x)) = f(x + 4) = (x + 4) - 4 = x
\]

Calculating \( g(f(x)) \):
\[
g(f(x)) = g(x - 4) = (x - 4) + 4 = x
\]

Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), **\( f(x) \) and \( g(x) \) are inverse functions**.

---

### b.
- \( f(x) = x - 4 \)
- \( g(x) = \frac{x - 4}{4} \)

Calculating \( f(g(x)) \):
\[
f(g(x)) = f\left(\frac{x - 4}{4}\right) = \frac{x - 4}{4} - 4 = \frac{x - 4 - 16}{4} = \frac{x - 20}{4}
\]
This is not equal to \( x \).

Calculating \( g(f(x)) \):
\[
g(f(x)) = g(x - 4) = \frac{(x - 4) - 4}{4} = \frac{x - 8}{4}
\]
This is also not equal to \( x \).

Thus, **\( f(x) \) and \( g(x) \) are not inverse functions**.

---

### c.
- \( f(x) = x - 4 \)
- \( g(x) = 4x - 1 \)

Calculating \( f(g(x)) \):
\[
f(g(x)) = f(4x - 1) = (4x - 1) - 4 = 4x - 5
\]
This is not equal to \( x \).

Calculating \( g(f(x)) \):
\[
g(f(x)) = g(x - 4) = 4(x - 4) - 1 = 4x - 16 - 1 = 4x - 17
\]
This is also not equal to \( x \).

Thus, **\( f(x) \) and \( g(x) \) are not inverse functions**.

### Conclusion:
The only pair of functions that are inverse functions is:
**a. \( f(x) = x - 4 \) and \( g(x) = x + 4 \)**.