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Let's solve the two problems step by step:

### 7. Find the inverse of \( f(x) = \frac{3x - 4}{2x + 1} \). And show that \( (f \circ f^{-1})(x) = x \).

#### Step 1: Find the inverse \( f^{-1}(x) \)
Start with the equation for \( f(x) \):
\[
y = \frac{3x - 4}{2x + 1}
\]
To find the inverse, solve for \( x \) in terms of \( y \).

1. Multiply both sides by \( 2x + 1 \) to get rid of the denominator:
   \[
   y(2x + 1) = 3x - 4
   \]
2. Expand the left side:
   \[
   2xy + y = 3x - 4
   \]
3. Collect all terms involving \( x \) on one side:
   \[
   2xy - 3x = -y - 4
   \]
4. Factor out \( x \):
   \[
   x(2y - 3) = -y - 4
   \]
5. Solve for \( x \):
   \[
   x = \frac{-y - 4}{2y - 3}
   \]
So, the inverse function is:
\[
f^{-1}(x) = \frac{-x - 4}{2x - 3}
\]

#### Step 2: Show that \( (f \circ f^{-1})(x) = x \)
We need to verify that applying \( f \) to its inverse returns \( x \).

Let’s compute \( f(f^{-1}(x)) \):
\[
f(f^{-1}(x)) = f\left( \frac{-x - 4}{2x - 3} \right)
\]
Substitute \( f^{-1}(x) \) into the original function \( f(x) = \frac{3x - 4}{2x + 1} \):
\[
f\left( \frac{-x - 4}{2x - 3} \right) = \frac{3\left( \frac{-x - 4}{2x - 3} \right) - 4}{2\left( \frac{-x - 4}{2x - 3} \right) + 1}
\]
Simplify the expression. After simplification, it should result in \( x \), proving that \( (f \circ f^{-1})(x) = x \).

---

### 8. Find the inverse of \( f(x) = 4x + 3 \). And show that \( (f \circ f^{-1})(x) = x \).

#### Step 1: Find the inverse \( f^{-1}(x) \)
Start with:
\[
y = 4x + 3
\]
To find the inverse, solve for \( x \) in terms of \( y \).

1. Subtract 3 from both sides:
   \[
   y - 3 = 4x
   \]
2. Divide by 4:
   \[
   x = \frac{y - 3}{4}
   \]
So, the inverse function is:
\[
f^{-1}(x) = \frac{x - 3}{4}
\]

#### Step 2: Show that \( (f \circ f^{-1})(x) = x \)
We need to verify that applying \( f \) to its inverse returns \( x \).

Let’s compute \( f(f^{-1}(x)) \):
\[
f(f^{-1}(x)) = f\left( \frac{x - 3}{4} \right)
\]
Substitute \( f^{-1}(x) \) into the original function \( f(x) = 4x + 3 \):
\[
f\left( \frac{x - 3}{4} \right) = 4\left( \frac{x - 3}{4} \right) + 3 = (x - 3) + 3 = x
\]
This shows that \( (f \circ f^{-1})(x) = x \).

---

Would you like more details or have any questions? Here are five related questions you might find useful:

1. How do you find the domain of an inverse function?
2. What are the graphical properties of a function and its inverse?
3. Can all functions have inverses? Why or why not?
4. How does the horizontal line test help in determining if a function is invertible?
5. What is the relationship between the composition of functions and identity functions?

**Tip:** When finding the inverse of a rational function, always remember to switch \( x \) and \( y \) and solve for \( y \) step by step, carefully handling algebraic expressions.

Find the inverse of f(x) = (3x - 4)/(2x + 1). And show that (f o f^(-1))(x) = x. Find the inverse of f(x) = 4x + 3. And show that (f o f^(-1))(x) = x.

Learn how to find the inverse of rational and linear functions such as f(x) = (3x - 4)/(2x + 1) and f(x) = 4x + 3. This problem also demonstrates how to prove that (f o f^(-1))(x) = x through function composition. Suitable for Grades 10-12.

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