EXERCISE 2 Consider the following functions: $$ \begin{array}{lll} ext { (a) } f(x)=2 x+4 & ext { (b) } f(x)=3 x-6 & ext { (c) } f(x)=\frac{1}{2} x+3 \ ext { (d) } f(x)=4 x-1 & ext { (e) } f(x)=-2 x & ext { (f) } f(x)=2 x-5\end{array} $$ $$ \begin{array}{ll} ext { (1) For each function determine } f^{-1} ext {, the inverse function in the form } f^{-1}(x)=\ldots \ldots \ ext { (2) Hence draw neat sketch graphs of both functions on the same set of axes. } \ ext { (3) Draw the line of symmetry on the same set of axes as the two graphs. } \ ext { (4) Determine the coordinates of the point of intersection of both graphs and } \ ext { then indicate this point on the diagram. }\end{array} $$

Question

EXERCISE 2 Consider the following functions: $$ \begin{array}{lll}	ext { (a) } f(x)=2 x+4 & 	ext { (b) } f(x)=3 x-6 & 	ext { (c) } f(x)=\frac{1}{2} x+3 \ 	ext { (d) } f(x)=4 x-1 & 	ext { (e) } f(x)=-2 x & 	ext { (f) } f(x)=2 x-5\end{array} $$ $$ \begin{array}{ll}	ext { (1) For each function determine } f^{-1} 	ext {, the inverse function in the form } f^{-1}(x)=\ldots \ldots \ 	ext { (2) Hence draw neat sketch graphs of both functions on the same set of axes. } \ 	ext { (3) Draw the line of symmetry on the same set of axes as the two graphs. } \ 	ext { (4) Determine the coordinates of the point of intersection of both graphs and } \ 	ext { then indicate this point on the diagram. }\end{array} $$

UpStudy AI · Accepted Answer

Let's solve the problem step by step for each function.

### Step 1: Find the Inverse Functions

To find the inverse function $$ f^{-1}(x) $$ for each function, we will follow these steps:
1. Replace $$ f(x) $$ with $$ y $$.
2. Solve for $$ x $$ in terms of $$ y $$.
3. Swap $$ x $$ and $$ y $$ to get the inverse function.

#### (a) $$ f(x) = 2x + 4 $$

1. $$ y = 2x + 4 $$
2. Solve for $$ x $$:
   $$
   y - 4 = 2x \implies x = \frac{y - 4}{2}
   $$
3. Swap $$ x $$ and $$ y $$:
   $$
   f^{-1}(x) = \frac{x - 4}{2}
   $$

#### (b) $$ f(x) = 3x - 6 $$

1. $$ y = 3x - 6 $$
2. Solve for $$ x $$:
   $$
   y + 6 = 3x \implies x = \frac{y + 6}{3}
   $$
3. Swap $$ x $$ and $$ y $$:
   $$
   f^{-1}(x) = \frac{x + 6}{3}
   $$

#### (c) $$ f(x) = \frac{1}{2}x + 3 $$

1. $$ y = \frac{1}{2}x + 3 $$
2. Solve for $$ x $$:
   $$
   y - 3 = \frac{1}{2}x \implies x = 2(y - 3)
   $$
3. Swap $$ x $$ and $$ y $$:
   $$
   f^{-1}(x) = 2(x - 3) = 2x - 6
   $$

#### (d) $$ f(x) = 4x - 1 $$

1. $$ y = 4x - 1 $$
2. Solve for $$ x $$:
   $$
   y + 1 = 4x \implies x = \frac{y + 1}{4}
   $$
3. Swap $$ x $$ and $$ y $$:
   $$
   f^{-1}(x) = \frac{x + 1}{4}
   $$

#### (e) $$ f(x) = -2x $$

1. $$ y = -2x $$
2. Solve for $$ x $$:
   $$
   y = -2x \implies x = -\frac{y}{2}
   $$
3. Swap $$ x $$ and $$ y $$:
   $$
   f^{-1}(x) = -\frac{x}{2}
   $$

#### (f) $$ f(x) = 2x - 5 $$

1. $$ y = 2x - 5 $$
2. Solve for $$ x $$:
   $$
   y + 5 = 2x \implies x = \frac{y + 5}{2}
   $$
3. Swap $$ x $$ and $$ y $$:
   $$
   f^{-1}(x) = \frac{x + 5}{2}
   $$

### Summary of Inverse Functions

- (a) $$ f^{-1}(x) = \frac{x - 4}{2} $$
- (b) $$ f^{-1}(x) = \frac{x + 6}{3} $$
- (c) $$ f^{-1}(x) = 2x - 6 $$
- (d) $$ f^{-1}(x) = \frac{x + 1}{4} $$
- (e) $$ f^{-1}(x) = -\frac{x}{2} $$
- (f) $$ f^{-1}(x) = \frac{x + 5}{2} $$

### Step 2: Sketch Graphs

To sketch the graphs of both functions and their inverses, we will plot the original functions and their inverses on the same axes. The line of symmetry for each function and its inverse is $$ y = x $$.

### Step 3: Draw the Line of Symmetry

The line of symmetry is $$ y = x $$. This line will be drawn on the same axes as the functions.

### Step 4: Determine the Coordinates of the Point of Intersection

To find the intersection points of each function and its inverse, we will set $$ f(x) = f^{-1}(x) $$ for each pair.

Let's calculate the intersection points for each function and its inverse.

#### (a) $$ 2x + 4 = \frac{x - 4}{2} $$

1. Multiply both sides by 2:
   $$
   4x + 8 = x - 4
   $$
2. Rearranging gives:
   $$
   3x = -12 \implies x = -4
   $$
3. Substitute $$ x = -4 $$ into $$ f(x) $$:
   $$
   f(-4) = 2(-4) + 4 = -8 + 4 = -4
   $$
4. Intersection point: $$ (-4, -4) $$

#### (b) $$ 3x - 6 = \frac{x + 6}{3} $$

1. Multiply both sides by 3:
   $$
   9x - 18 = x + 6
   $$
2. Rearranging gives:
   $$
   8x = 24 \implies x = 3
   $$
3. Substitute $$ x = 3 $$ into $$ f(x) $$:
   $$
   f(3) = 3(3) - 6 = 9 - 6 = 3
   $$
4. Intersection point: $$ (3, 3) $$

#### (c) $$ \frac{1}{2}x + 3 = 2x - 6 $$

1. Multiply both sides by 2:
   $$
   x + 6 = 4x - 12
   $$
2. Rearranging gives:
   $$
   3x = 18 \implies x = 6
   $$
3. Substitute $$ x = 6 $$ into $$ f(x) $$:
   $$
   f(6) = \frac{1}{2}(6) + 3 = 3 + 3 = 6
   $$
4. Intersection point: $$ (6, 6) $$

#### (d) $$ 4x - 1 = \frac{x + 1}{4} $$

1. Multiply both sides by 4:
   $$
   16x - 4 = x + 1
   $$
2. Rearranging gives:
   \[
   15x = 5 \im
Here are the inverse functions and the coordinates of their intersection points: - **(a)** - Inverse Function: $$ f^{-1}(x) = \frac{x - 4}{2} $$ - Intersection Point: $$ (-4, -4) $$ - **(b)** - Inverse Function: $$ f^{-1}(x) = \frac{x + 6}{3} $$ - Intersection Point: $$ (3, 3) $$ - **(c)** - Inverse Function: $$ f^{-1}(x) = 2x - 6 $$ - Intersection Point: $$ (6, 6) $$ - **(d)** - Inverse Function: $$ f^{-1}(x) = \frac{x + 1}{4} $$ - Intersection Point: $$ \left(\frac{5}{15}, \frac{5}{15}\right) = \left(\frac{1}{3}, \frac{1}{3}\right) $$ - **(e)** - Inverse Function: $$ f^{-1}(x) = -\frac{x}{2} $$ - Intersection Point: $$ (0, 0) $$ - **(f)** - Inverse Function: $$ f^{-1}(x) = \frac{x + 5}{2} $$ - Intersection Point: $$ (-5, -5) $$ **Summary:** - Each function has an inverse function expressed in terms of $$ x $$. - The intersection points of each function with its inverse lie on the line $$ y = x $$. - The coordinates of these intersection points are as listed above.

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