Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Absolutely, let's go through the process step-by-step.
### Step 1: Understanding the Function [tex]\( y = |x| + 4 \)[/tex]
The function [tex]\( y = |x| + 4 \)[/tex] involves the absolute value of [tex]\( x \)[/tex], which means it takes different expressions for [tex]\( x \)[/tex] values less than 0 and greater than or equal to 0:
1. When [tex]\( x \geq 0 \)[/tex], [tex]\( |x| = x \)[/tex], so the function becomes [tex]\( y = x + 4 \)[/tex].
2. When [tex]\( x < 0 \)[/tex], [tex]\( |x| = -x \)[/tex], so the function becomes [tex]\( y = -x + 4 \)[/tex].
### Step 2: Plotting Points for [tex]\( y = |x| + 4 \)[/tex]
We need to plot points for both cases:
For [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x = 0 \implies y = 0 + 4 = 4 \)[/tex]
- [tex]\( x = 1 \implies y = 1 + 4 = 5 \)[/tex]
- [tex]\( x = 2 \implies y = 2 + 4 = 6 \)[/tex]
- [tex]\( x = 3 \implies y = 3 + 4 = 7 \)[/tex]
- [tex]\( x = 4 \implies y = 4 + 4 = 8 \)[/tex]
For [tex]\( x < 0 \)[/tex]:
- [tex]\( x = -1 \implies y = -(-1) + 4 = 5 \)[/tex]
- [tex]\( x = -2 \implies y = -(-2) + 4 = 6 \)[/tex]
- [tex]\( x = -3 \implies y = -(-3) + 4 = 7 \)[/tex]
- [tex]\( x = -4 \implies y = -(-4) + 4 = 8 \)[/tex]
- [tex]\( x = -5 \implies y = -(-5) + 4 = 9 \)[/tex]
### Step 3: Plot these Points
Create a plot with the [tex]\( x \)[/tex] and corresponding [tex]\( y \)[/tex] values.
### Step 4: Reflect Across the Line [tex]\( y = x \)[/tex]
To obtain the inverse function, we can reflect the graph across the line [tex]\( y = x \)[/tex]. Reflecting over [tex]\( y = x \)[/tex] involves swapping each [tex]\( (x, y) \)[/tex] coordinate to [tex]\( (y, x) \)[/tex].
Points after reflection:
- [tex]\( (0, 4) \rightarrow (4, 0) \)[/tex]
- [tex]\( (1, 5) \rightarrow (5, 1) \)[/tex]
- [tex]\( (2, 6) \rightarrow (6, 2) \)[/tex]
- [tex]\( (3, 7) \rightarrow (7, 3) \)[/tex]
- [tex]\( (4, 8) \rightarrow (8, 4) \)[/tex]
- [tex]\( (5, 9) \rightarrow (9, 5) \)[/tex]
- [tex]\( (-1, 5) \rightarrow (5, -1) \)[/tex]
- [tex]\( (-2, 6) \rightarrow (6, -2) \)[/tex]
- [tex]\( (-3, 7) \rightarrow (7, -3) \)[/tex]
- [tex]\( (-4, 8) \rightarrow (8, -4) \)[/tex]
- [tex]\( (-5, 9) \rightarrow (9, -5) \)[/tex]
These are now the points of the inverse function, and you can plot these on the graph.
### Step 5: Graphing
Using graph paper or graphing software, perform the following:
1. Plot the points for [tex]\( y = |x| + 4 \)[/tex].
2. Draw the line [tex]\( y = x \)[/tex] as a reference.
3. Plot the reflected points (inverse function).
The resulting graph will show the original function [tex]\( y = |x| + 4 \)[/tex] and its inverse reflected across the line [tex]\( y = x \)[/tex].
### Final Graph
1. The original function [tex]\( y = |x| + 4 \)[/tex] forms a V-shape starting from [tex]\( (0, 4) \)[/tex].
2. The inverse graph will be a reflected V-shape, showing points like [tex]\( (4, 0) \)[/tex], [tex]\( (5, 1) \)[/tex], and so on.
This methodological approach enables us to graph the function and its inverse manually without computational errors.
### Step 1: Understanding the Function [tex]\( y = |x| + 4 \)[/tex]
The function [tex]\( y = |x| + 4 \)[/tex] involves the absolute value of [tex]\( x \)[/tex], which means it takes different expressions for [tex]\( x \)[/tex] values less than 0 and greater than or equal to 0:
1. When [tex]\( x \geq 0 \)[/tex], [tex]\( |x| = x \)[/tex], so the function becomes [tex]\( y = x + 4 \)[/tex].
2. When [tex]\( x < 0 \)[/tex], [tex]\( |x| = -x \)[/tex], so the function becomes [tex]\( y = -x + 4 \)[/tex].
### Step 2: Plotting Points for [tex]\( y = |x| + 4 \)[/tex]
We need to plot points for both cases:
For [tex]\( x \geq 0 \)[/tex]:
- [tex]\( x = 0 \implies y = 0 + 4 = 4 \)[/tex]
- [tex]\( x = 1 \implies y = 1 + 4 = 5 \)[/tex]
- [tex]\( x = 2 \implies y = 2 + 4 = 6 \)[/tex]
- [tex]\( x = 3 \implies y = 3 + 4 = 7 \)[/tex]
- [tex]\( x = 4 \implies y = 4 + 4 = 8 \)[/tex]
For [tex]\( x < 0 \)[/tex]:
- [tex]\( x = -1 \implies y = -(-1) + 4 = 5 \)[/tex]
- [tex]\( x = -2 \implies y = -(-2) + 4 = 6 \)[/tex]
- [tex]\( x = -3 \implies y = -(-3) + 4 = 7 \)[/tex]
- [tex]\( x = -4 \implies y = -(-4) + 4 = 8 \)[/tex]
- [tex]\( x = -5 \implies y = -(-5) + 4 = 9 \)[/tex]
### Step 3: Plot these Points
Create a plot with the [tex]\( x \)[/tex] and corresponding [tex]\( y \)[/tex] values.
### Step 4: Reflect Across the Line [tex]\( y = x \)[/tex]
To obtain the inverse function, we can reflect the graph across the line [tex]\( y = x \)[/tex]. Reflecting over [tex]\( y = x \)[/tex] involves swapping each [tex]\( (x, y) \)[/tex] coordinate to [tex]\( (y, x) \)[/tex].
Points after reflection:
- [tex]\( (0, 4) \rightarrow (4, 0) \)[/tex]
- [tex]\( (1, 5) \rightarrow (5, 1) \)[/tex]
- [tex]\( (2, 6) \rightarrow (6, 2) \)[/tex]
- [tex]\( (3, 7) \rightarrow (7, 3) \)[/tex]
- [tex]\( (4, 8) \rightarrow (8, 4) \)[/tex]
- [tex]\( (5, 9) \rightarrow (9, 5) \)[/tex]
- [tex]\( (-1, 5) \rightarrow (5, -1) \)[/tex]
- [tex]\( (-2, 6) \rightarrow (6, -2) \)[/tex]
- [tex]\( (-3, 7) \rightarrow (7, -3) \)[/tex]
- [tex]\( (-4, 8) \rightarrow (8, -4) \)[/tex]
- [tex]\( (-5, 9) \rightarrow (9, -5) \)[/tex]
These are now the points of the inverse function, and you can plot these on the graph.
### Step 5: Graphing
Using graph paper or graphing software, perform the following:
1. Plot the points for [tex]\( y = |x| + 4 \)[/tex].
2. Draw the line [tex]\( y = x \)[/tex] as a reference.
3. Plot the reflected points (inverse function).
The resulting graph will show the original function [tex]\( y = |x| + 4 \)[/tex] and its inverse reflected across the line [tex]\( y = x \)[/tex].
### Final Graph
1. The original function [tex]\( y = |x| + 4 \)[/tex] forms a V-shape starting from [tex]\( (0, 4) \)[/tex].
2. The inverse graph will be a reflected V-shape, showing points like [tex]\( (4, 0) \)[/tex], [tex]\( (5, 1) \)[/tex], and so on.
This methodological approach enables us to graph the function and its inverse manually without computational errors.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
