determine whether f and g are inverses function 

f(×)=3×+1.       g(×)=×-1/3

To determine whether \( f(x) \) and \( g(x) \) are inverse functions, we need to verify if:

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

Let’s start with the first condition:

### Step 1: Check if \( f(g(x)) = x \)

Substitute \( g(x) = \frac{x - 1}{3} \) into \( f(x) = 3x + 1 \):

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

Simplifying this:

\[
f(g(x)) = (x - 1) + 1 = x
\]

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

### Step 2: Check if \( g(f(x)) = x \)

Substitute \( f(x) = 3x + 1 \) into \( g(x) = \frac{x - 1}{3} \):

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

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

### Conclusion:
Since both \( f(g(x)) = x \) and \( g(f(x)) = x \), the functions \( f(x) = 3x + 1 \) and \( g(x) = \frac{x - 1}{3} \) are indeed inverses of each other.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the inverse of more complex functions?
2. What happens if only one of \( f(g(x)) = x \) or \( g(f(x)) = x \) holds?
3. How can you graphically verify that two functions are inverses?
4. Can inverse functions exist for non-linear functions?
5. How do you handle finding inverses when dealing with trigonometric functions?

**Tip:** The graph of a function and its inverse are reflections over the line \( y = x \).

determine whether f and g are inverse functions

f(x) = 3x + 1
g(x) = (x - 1) / 3

Explore how to determine if f(x) = 3x + 1 and g(x) = (x - 1)/3 are inverse functions. This problem involves key algebraic concepts such as function composition and the theorem of inverse functions. Step-by-step solution provided for students in Grades 10-12.

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