Given the function $$ f(x)=4 x+4 $$. (1) Determine the equation of $$ f^{-1} $$. (2) Sketch the graphs of $$ f $$ and $$ f^{-1} $$ on the same set of axes and show the line of symmetry. (3) Calculate the coordinates of the point of intersection between $$ f $$ and $$ f^{-1} $$.

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Let's address each part of the problem step-by-step.

### **1. Determining the Inverse Function $$ f^{-1}(x) $$**

Given the function:
$$ f(x) = 4x + 4 $$

To find the inverse function $$ f^{-1}(x) $$, follow these steps:

1. **Replace $$ f(x) $$ with $$ y $$:**
   $$ y = 4x + 4 $$

2. **Solve for $$ x $$:**
   $$
   \begin{align*}
   y &= 4x + 4 \
   y - 4 &= 4x \
   x &= \frac{y - 4}{4}
   \end{align*}
   $$

3. **Express the inverse function:**
   $$ f^{-1}(x) = \frac{x - 4}{4} $$

### **2. Sketching the Graphs of $$ f $$ and $$ f^{-1} $$ and the Line of Symmetry**

- **Graph of $$ f(x) = 4x + 4 $$:**
  - **Slope:** 4
  - **Y-intercept:** (0, 4)
  - **Example Points:**
    - When $$ x = 0 $$, $$ y = 4 $$
    - When $$ x = 1 $$, $$ y = 8 $$

- **Graph of $$ f^{-1}(x) = \frac{x - 4}{4} $$:**
  - **Slope:** $$ \frac{1}{4} $$
  - **Y-intercept:** When $$ x = 0 $$, $$ y = -1 $$ ⇒ (0, -1)
  - **Example Points:**
    - When $$ x = 4 $$, $$ y = 0 $$
    - When $$ x = 8 $$, $$ y = 1 $$

- **Line of Symmetry $$ y = x $$:**
  - This is a 45-degree line passing through the origin.
  - Both $$ f $$ and $$ f^{-1} $$ are symmetric with respect to this line.

**Visualization:**
1. Draw the line $$ y = x $$.
2. Plot $$ f(x) = 4x + 4 $$ with a steep slope, crossing the y-axis at (0, 4).
3. Plot $$ f^{-1}(x) = \frac{x - 4}{4} $$ with a gentler slope, crossing the y-axis at (0, -1) and the x-axis at (4, 0).
4. Notice that both lines intersect on the line $$ y = x $$, illustrating their symmetry.

### **3. Calculating the Intersection Point Between $$ f $$ and $$ f^{-1} $$**

To find the intersection, set $$ f(x) $$ equal to $$ f^{-1}(x) $$:
$$
\begin{align*}
4x + 4 &= \frac{x - 4}{4} \
\text{Multiply both sides by 4 to eliminate the fraction:} \
16x + 16 &= x - 4 \
\text{Rearrange terms:} \
16x - x &= -4 - 16 \
15x &= -20 \
x &= -\frac{20}{15} = -\frac{4}{3}
\end{align*}
$$

Now, find $$ y $$:
$$
y = f(x) = 4\left(-\frac{4}{3}\right) + 4 = -\frac{16}{3} + 4 = -\frac{16}{3} + \frac{12}{3} = -\frac{4}{3}
$$

**Intersection Point:**
$$
\left( -\frac{4}{3},\ -\frac{4}{3} \right)
$$

This point lies on the line $$ y = x $$, confirming the symmetry.

### **Summary**

1. **Inverse Function:**  
   $$ f^{-1}(x) = \frac{x - 4}{4} $$

2. **Graphs and Symmetry:**  
   - $$ f(x) = 4x + 4 $$ and $$ f^{-1}(x) = \frac{x - 4}{4} $$ are straight lines.
   - They are symmetric with respect to the line $$ y = x $$.

3. **Intersection Point:**  
   $$ \left( -\frac{4}{3},\ -\frac{4}{3} \right) $$
1. **Inverse Function:** $$ f^{-1}(x) = \frac{x - 4}{4} $$ 2. **Graphs and Symmetry:** - Both $$ f(x) = 4x + 4 $$ and $$ f^{-1}(x) = \frac{x - 4}{4} $$ are straight lines. - They are symmetric with respect to the line $$ y = x $$. 3. **Intersection Point:** $$ \left( -\frac{4}{3},\ -\frac{4}{3} \right) $$

Given the function \( f(x)=4 x+4 \). (1) Determine the equation of \( f^{-1} \). (2) Sketch the graphs of \( f \) and \( f^{-1} \) on the same set of axes and show the line of symmetry. (3) Calculate the coordinates of the point of intersection between \( f \) and \( f^{-1} \).

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