Graph the given function using the given values of x. Also use the graph to determine the domain and range of the function.
1. f(x)=x^2-1 ; x = -3, -2, -1, 0, 1, 2, 3
2. g(x)=√(x+1)-2 ; x = -1, 0, 3, 8
3. h(x)=|x-3| ; x = 0, 1, 2, 3, 4, 5, 6
To graph the given functions and determine the domain and range, follow these steps:
1. f(x) = x^2 - 1 ; x = -3, -2, -1, 0, 1, 2, 3
To graph f(x), substitute the given values of x into the equation and calculate the corresponding values of f(x):
- When x = -3: f(-3) = (-3)^2 - 1 = 9 - 1 = 8
- When x = -2: f(-2) = (-2)^2 - 1 = 4 - 1 = 3
- When x = -1: f(-1) = (-1)^2 - 1 = 1 - 1 = 0
- When x = 0: f(0) = (0)^2 - 1 = 0 - 1 = -1
- When x = 1: f(1) = (1)^2 - 1 = 1 - 1 = 0
- When x = 2: f(2) = (2)^2 - 1 = 4 - 1 = 3
- When x = 3: f(3) = (3)^2 - 1 = 9 - 1 = 8
The points to plot on the graph are (-3, 8), (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3), and (3, 8).
Now, plot these points on a graph:
|
10 | *
| * *
|* *
|____________________
-3 -2 -1 0 1 2 3
The graph of f(x) is a U-shaped curve called a parabola, opening upwards. Its vertex is at (0, -1).
The domain of the function is all real numbers since there are no restrictions on x.
The range of the function is values of f(x) as x ranges over its domain. From the graph, you can see that the function goes from -1 to positive infinity, so the range is [-1, ∞).
2. g(x) = √(x + 1) - 2 ; x = -1, 0, 3, 8
Substitute the given values of x into the equation to find the corresponding values of g(x):
- When x = -1: g(-1) = √(-1 + 1) - 2 = √0 - 2 = -2
- When x = 0: g(0) = √(0 + 1) - 2 = √1 - 2 = -1
- When x = 3: g(3) = √(3 + 1) - 2 = √4 - 2 = 0
- When x = 8: g(8) = √(8 + 1) - 2 = √9 - 2 = 1
Plot the points (-1, -2), (0, -1), (3, 0), and (8, 1) on a graph:
|
2 |
|
1 |
| *
| *
0 | *
|____________________
-1 0 1 2 3 4 5 6 7 8 9
The graph of g(x) is a curve that starts from a point above the line y = -2, then approaches y = -1, then crosses the x-axis at y = 0, and finally reaches y = 1 as x gets larger.
The domain of g(x) is all real numbers greater than or equal to -1 since the square root function is defined for non-negative values.
The range of g(x) is the set of values of g(x) as x ranges over its domain. From the graph, you can see that the range is approximately [-2, 1).
3. h(x) = |x - 3| ; x = 0, 1, 2, 3, 4, 5, 6
Substitute the given values of x into the equation to find the corresponding values of h(x):
- When x = 0: h(0) = |0 - 3| = |-3| = 3
- When x = 1: h(1) = |1 - 3| = |-2| = 2
- When x = 2: h(2) = |2 - 3| = |-1| = 1
- When x = 3: h(3) = |3 - 3| = |0| = 0
- When x = 4: h(4) = |4 - 3| = |1| = 1
- When x = 5: h(5) = |5 - 3| = |2| = 2
- When x = 6: h(6) = |6 - 3| = |3| = 3
Plot the points (0, 3), (1, 2), (2, 1), (3, 0), (4, 1), (5, 2), and (6, 3) on a graph:
|
4 |
|
3 | * *
| * *
2 | * *
|____________________________
0 1 2 3 4 5 6 7
The graph of h(x) consists of two V-shaped curves, symmetric about x = 3. It represents the absolute value function.
The domain of h(x) is all real numbers since there are no restrictions on x.
The range of h(x) is the set of values of h(x) as x ranges over its domain. From the graph, you can see that the range is [0, 3].