Inverse Functions worksheet with problems to find inverses using the flow-diagram method.

To solve the problem of finding the inverses of the given functions using the flow-diagram method, we will follow these steps:

1. Understand the function: Write down the original function.
2. Express \( y \) in terms of \( x \): Replace \( f(x) \) or \( h(x) \) with \( y \).
3. Solve for \( x \) in terms of \( y \): Rearrange the equation to isolate \( x \).
4. Replace \( y \) with \( x \): This gives the inverse function \( f^{-1}(x) \) or \( h^{-1}(x) \).

Let's solve each function step by step.

---

Problem 1: \( f(x) = 5x - 2 \)

#### Step 1: Write the function
\[ y = 5x - 2 \]

#### Step 2: Solve for \( x \)
\[ y = 5x - 2 \]
\[ y + 2 = 5x \]
\[ x = \frac{y + 2}{5} \]

#### Step 3: Replace \( y \) with \( x \)
\[ f^{-1}(x) = \frac{x + 2}{5} \]

---

Problem 2: \( h(x) = 5(x - 2) \)

#### Step 1: Write the function
\[ y = 5(x - 2) \]

#### Step 2: Solve for \( x \)
\[ y = 5(x - 2) \]
\[ \frac{y}{5} = x - 2 \]
\[ x = \frac{y}{5} + 2 \]

#### Step 3: Replace \( y \) with \( x \)
\[ h^{-1}(x) = \frac{x}{5} + 2 \]

---

Problem 3: \( l(x) = \frac{4 - x}{3} + 2 \)

#### Step 1: Write the function
\[ y = \frac{4 - x}{3} + 2 \]

#### Step 2: Solve for \( x \)
\[ y = \frac{4 - x}{3} + 2 \]
\[ y - 2 = \frac{4 - x}{3} \]
\[ 3(y - 2) = 4 - x \]
\[ 3y - 6 = 4 - x \]
\[ x = 4 - 3y + 6 \]
\[ x = 10 - 3y \]

#### Step 3: Replace \( y \) with \( x \)
\[ l^{-1}(x) = 10 - 3x \]

---

Problem 4: \( g(x) = \frac{2x + 1}{3} \)

#### Step 1: Write the function
\[ y = \frac{2x + 1}{3} \]

#### Step 2: Solve for \( x \)
\[ y = \frac{2x + 1}{3} \]
\[ 3y = 2x + 1 \]
\[ 3y - 1 = 2x \]
\[ x = \frac{3y - 1}{2} \]

#### Step 3: Replace \( y \) with \( x \)
\[ g^{-1}(x) = \frac{3x - 1}{2} \]

---

Problem 5: \( h(x) = 3(2x + 4) \)

#### Step 1: Write the function
\[ y = 3(2x + 4) \]

#### Step 2: Solve for \( x \)
\[ y = 3(2x + 4) \]
\[ \frac{y}{3} = 2x + 4 \]
\[ \frac{y}{3} - 4 = 2x \]
\[ x = \frac{\frac{y}{3} - 4}{2} \]
\[ x = \frac{y}{6} - 2 \]

#### Step 3: Replace \( y \) with \( x \)
\[ h^{-1}(x) = \frac{x}{6} - 2 \]

---

Problem 6: \( k(x) = -7x + 3 \)

#### Step 1: Write the function
\[ y = -7x + 3 \]

#### Step 2: Solve for \( x \)
\[ y = -7x + 3 \]
\[ y - 3 = -7x \]
\[ x = \frac{y - 3}{-7} \]
\[ x = -\frac{y - 3}{7} \]

#### Step 3: Replace \( y \) with \( x \)
\[ k^{-1}(x) = -\frac{x - 3}{7} \]

---

Problem 7: \( m(x) = 2(3x + 4) - 6 \)

#### Step 1: Write the function
\[ y = 2(3x + 4) - 6 \]

#### Step 2: Solve for \( x \)
\[ y = 2(3x + 4) - 6 \]
\[ y = 6x + 8 - 6 \]
\[ y = 6x + 2 \]
\[ y - 2 = 6x \]
\[ x = \frac{y - 2}{6} \]

#### Step 3: Replace \( y \) with \( x \)
\[ m^{-1}(x) = \frac{x - 2}{6} \]

---

Final Answers

1. \( f^{-1}(x) = \frac{x + 2}{5} \)
2. \( h^{-1}(x) = \frac{x}{5} + 2 \)
3. \( l^{-1}(x) = 10 - 3x \)
4. \( g^{-1}(x) = \frac{3x - 1}{2} \)
5. \( h^{-1}(x) = \frac{x}{6} - 2 \)
6. \( k^{-1}(x) = -\frac{x - 3}{7} \)
7. \( m^{-1}(x) = \frac{x - 2}{6} \)

\[
\boxed{
\begin{aligned}
1. & \quad f^{-1}(x) = \frac{x + 2}{5} \\
2. & \quad h^{-1}(x) = \frac{x}{5} + 2 \\
3. & \quad l^{-1}(x) = 10 - 3x \\
4. & \quad g^{-1}(x) = \frac{3x - 1}{2} \\
5. & \quad h^{-1}(x) = \frac{x}{6} - 2 \\
6. & \quad k^{-1}(x) = -\frac{x - 3}{7} \\
7. & \quad m^{-1}(x) = \frac{x - 2}{6}
\end{aligned}
}
\]