https://file.aiquickdraw.com/mathbot/file/1730239955868hhtdozoc.jpg

Let's solve each part of the question step by step. We have two functions defined as follows:

\[
f(x) = -5x + 3
\]
\[
g(x) = \frac{1}{5}(x - 3)
\]

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

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{1}{5}(x - 3)\right)
\]
Now, substitute this expression into \( f(x) = -5x + 3 \):
\[
f\left(\frac{1}{5}(x - 3)\right) = -5\left(\frac{1}{5}(x - 3)\right) + 3
\]
Simplify the expression:
\[
= -(x - 3) + 3
\]
\[
= -x + 3 + 3
\]
\[
= -x + 6
\]

So, \( f(g(x)) = -x + 6 \).

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

To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(-5x + 3)
\]
Now, substitute this expression into \( g(x) = \frac{1}{5}(x - 3) \):
\[
g(-5x + 3) = \frac{1}{5}((-5x + 3) - 3)
\]
Simplify the expression inside the parentheses:
\[
= \frac{1}{5}(-5x)
\]
\[
= -x
\]

So, \( g(f(x)) = -x \).

### (c) Determine whether the functions \( f \) and \( g \) are inverses of each other

Two functions \( f \) and \( g \) are inverses of each other if \( f(g(x)) = x \) and \( g(f(x)) = x \).

From parts (a) and (b), we found:
\[
f(g(x)) = -x + 6
\]
\[
g(f(x)) = -x
\]

Since neither \( f(g(x)) = x \) nor \( g(f(x)) = x \), the functions \( f \) and \( g \) are **not inverses of each other**.

Would you like more details or have questions about any steps?

### Further Questions
1. How do we find the inverse of a function in general?
2. What properties do inverse functions satisfy?
3. Can you explain the steps in simplifying \( f(g(x)) \)?
4. What is the difference between composition of functions and inverse functions?
5. How can we verify if a function is one-to-one?

### Tip
When testing if two functions are inverses, remember to check both \( f(g(x)) = x \) and \( g(f(x)) = x \); both conditions must be true for them to be inverses.

Consider the functions f(x) = -5x + 3 and g(x) = 1/5(x - 3). (a) Find f(g(x)). (b) Find g(f(x)). (c) Determine whether the functions f and g are inverses of each other.

Step-by-step solution for determining if f(x) = -5x + 3 and g(x) = 1/5(x - 3) are inverse functions. Learn to calculate f(g(x)) and g(f(x)) and apply the definition of inverse functions. Ideal for Grades 10-12 students exploring function composition.

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