Question

code: ______ name: hugo andre t. galang section: 9 - mdl 4 - dec 2025
graph the inverse for the given functions.
1)
2)

1. $f(x) = 3x + 3$

$x$	-1	0	1
$f(x)$	0	3	6

1. $g(x) = (x - 3)^2 - 1, x \leq 3$

$x$	3	2	1
$g(x)$	-1	0	3

Explanation:

To graph the inverse of a function, we use the property that the graph of the inverse function \( f^{-1}(x) \) is the reflection of the graph of \( f(x) \) over the line \( y = x \). We can also find the inverse function algebraically and then plot points or use the reflection method. Let's handle each part:

1. Graphing the Inverse of the First Function (Graphical)

• Step 1: Identify Key Points

For the given linear graph, find two or more points on \( f(x) \). For example, if the line passes through \( (0, 3) \) and \( (2, -1) \) (estimating from the grid), the inverse function’s points will be \( (3, 0) \) and \( (-1, 2) \) (swap \( x \) and \( y \)).

• Step 2: Reflect Over \( y = x \)

Plot the reflected points and draw the line through them. The inverse of a linear function is also linear, so the reflected line will be the inverse.

2. Graphing the Inverse of the Second Function (Graphical)

• Step 1: Identify Key Points

For the piecewise graph, find points on \( g(x) \) (e.g., \( (0, 1) \), \( (2, 2) \), \( (4, 5) \) – estimating from the grid). Swap \( x \) and \( y \) to get points for \( g^{-1}(x) \): \( (1, 0) \), \( (2, 2) \), \( (5, 4) \).

• Step 2: Reflect Over \( y = x \)

Plot these reflected points and draw the curve. The inverse of a curve (like a piecewise function) will be the reflection over \( y = x \).

3. Inverse of \( f(x) = 3x + 3 \)

• Step 1: Find the Inverse Algebraically

Let \( y = 3x + 3 \). Swap \( x \) and \( y \):

$$ x = 3y + 3 $$

• Step 2: Solve for \( y \)

Subtract 3: \( x - 3 = 3y \). Divide by 3:

$$ y = \frac{x - 3}{3} = \frac{1}{3}x - 1 $$

So, \( f^{-1}(x) = \frac{1}{3}x - 1 \).

• Step 3: Plot Points

Use the table for \( f(x) \):

• For \( x = -1 \), \( f(-1) = 0 \) → inverse point: \( (0, -1) \).
• For \( x = 0 \), \( f(0) = 3 \) → inverse point: \( (3, 0) \).
• For \( x = 1 \), \( f(1) = 6 \) → inverse point: \( (6, 1) \).

Plot these points \( (0, -1) \), \( (3, 0) \), \( (6, 1) \) and draw the line \( y = \frac{1}{3}x - 1 \).

4. Inverse of \( g(x) = (x - 3)^2 - 1 \), \( x \leq 3 \)

• Step 1: Find the Inverse Algebraically

Let \( y = (x - 3)^2 - 1 \), \( x \leq 3 \). Swap \( x \) and \( y \):

$$ x = (y - 3)^2 - 1 $$

• Step 2: Solve for \( y \)

Add 1: \( x + 1 = (y - 3)^2 \). Take square roots (note \( x \leq 3 \) implies \( y - 3 \leq 0 \), so we use the negative root):

$$ \sqrt{x + 1} = 3 - y \implies y = 3 - \sqrt{x + 1} $$

So, \( g^{-1}(x) = 3 - \sqrt{x + 1} \).

• Step 3: Plot Points

Use the table for \( g(x) \):

• For \( x = 3 \), \( g(3) = -1 \) → inverse point: \( (-1, 3) \).
• For \( x = 2 \), \( g(2) = 0 \) → inverse point: \( (0, 2) \).
• For \( x = 1 \), \( g(1) = 3 \) → inverse point: \( (3, 1) \).

Plot these points \( (-1, 3) \), \( (0, 2) \), \( (3, 1) \) and draw the curve \( y = 3 - \sqrt{x + 1} \).

Final Answer (Graphing Summary)

To graph the inverse of a function:

1. Reflect over \( y = x \) for graphical functions.
2. Find the inverse algebraically (swap \( x \) and \( y \), solve for \( y \)) for explicit functions.
3. Plot reflected points or use the algebraic inverse to draw the graph.

For \( f(x) = 3x + 3 \), the inverse is \( \boldsymbol{f^{-1}(x) = \frac{1}{3}x - 1} \).
For \( g(x) = (x - 3)^2 - 1 \) (\( x \leq 3 \)), the inverse is \( \boldsymbol{g^{-1}(x) = 3 - \sqrt{x + 1}} \).

(Note: For the graphical parts, follow the reflection over \( y = x \) and plot the reflected points.)

Answer:

To graph the inverse of a function, we use the property that the graph of the inverse function \( f^{-1}(x) \) is the reflection of the graph of \( f(x) \) over the line \( y = x \). We can also find the inverse function algebraically and then plot points or use the reflection method. Let's handle each part:

1. Graphing the Inverse of the First Function (Graphical)

• Step 1: Identify Key Points

For the given linear graph, find two or more points on \( f(x) \). For example, if the line passes through \( (0, 3) \) and \( (2, -1) \) (estimating from the grid), the inverse function’s points will be \( (3, 0) \) and \( (-1, 2) \) (swap \( x \) and \( y \)).

• Step 2: Reflect Over \( y = x \)

Plot the reflected points and draw the line through them. The inverse of a linear function is also linear, so the reflected line will be the inverse.

2. Graphing the Inverse of the Second Function (Graphical)

• Step 1: Identify Key Points

For the piecewise graph, find points on \( g(x) \) (e.g., \( (0, 1) \), \( (2, 2) \), \( (4, 5) \) – estimating from the grid). Swap \( x \) and \( y \) to get points for \( g^{-1}(x) \): \( (1, 0) \), \( (2, 2) \), \( (5, 4) \).

• Step 2: Reflect Over \( y = x \)

Plot these reflected points and draw the curve. The inverse of a curve (like a piecewise function) will be the reflection over \( y = x \).

3. Inverse of \( f(x) = 3x + 3 \)

• Step 1: Find the Inverse Algebraically

Let \( y = 3x + 3 \). Swap \( x \) and \( y \):

$$ x = 3y + 3 $$

• Step 2: Solve for \( y \)

Subtract 3: \( x - 3 = 3y \). Divide by 3:

$$ y = \frac{x - 3}{3} = \frac{1}{3}x - 1 $$

So, \( f^{-1}(x) = \frac{1}{3}x - 1 \).

• Step 3: Plot Points

Use the table for \( f(x) \):

• For \( x = -1 \), \( f(-1) = 0 \) → inverse point: \( (0, -1) \).
• For \( x = 0 \), \( f(0) = 3 \) → inverse point: \( (3, 0) \).
• For \( x = 1 \), \( f(1) = 6 \) → inverse point: \( (6, 1) \).

Plot these points \( (0, -1) \), \( (3, 0) \), \( (6, 1) \) and draw the line \( y = \frac{1}{3}x - 1 \).

4. Inverse of \( g(x) = (x - 3)^2 - 1 \), \( x \leq 3 \)

• Step 1: Find the Inverse Algebraically

Let \( y = (x - 3)^2 - 1 \), \( x \leq 3 \). Swap \( x \) and \( y \):

$$ x = (y - 3)^2 - 1 $$

• Step 2: Solve for \( y \)

Add 1: \( x + 1 = (y - 3)^2 \). Take square roots (note \( x \leq 3 \) implies \( y - 3 \leq 0 \), so we use the negative root):

$$ \sqrt{x + 1} = 3 - y \implies y = 3 - \sqrt{x + 1} $$

So, \( g^{-1}(x) = 3 - \sqrt{x + 1} \).

• Step 3: Plot Points

Use the table for \( g(x) \):

• For \( x = 3 \), \( g(3) = -1 \) → inverse point: \( (-1, 3) \).
• For \( x = 2 \), \( g(2) = 0 \) → inverse point: \( (0, 2) \).
• For \( x = 1 \), \( g(1) = 3 \) → inverse point: \( (3, 1) \).

Plot these points \( (-1, 3) \), \( (0, 2) \), \( (3, 1) \) and draw the curve \( y = 3 - \sqrt{x + 1} \).

Final Answer (Graphing Summary)

To graph the inverse of a function:

1. Reflect over \( y = x \) for graphical functions.
2. Find the inverse algebraically (swap \( x \) and \( y \), solve for \( y \)) for explicit functions.
3. Plot reflected points or use the algebraic inverse to draw the graph.

For \( f(x) = 3x + 3 \), the inverse is \( \boldsymbol{f^{-1}(x) = \frac{1}{3}x - 1} \).
For \( g(x) = (x - 3)^2 - 1 \) (\( x \leq 3 \)), the inverse is \( \boldsymbol{g^{-1}(x) = 3 - \sqrt{x + 1}} \).

(Note: For the graphical parts, follow the reflection over \( y = x \) and plot the reflected points.)