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 their domain and range, follow these steps:
1. For each value of x, calculate the corresponding value of y by substituting x into the given function.
2. Plot each point (x, y) on a coordinate plane.
3. Connect the points to create the graph of the function.
4. Analyze the graph to determine the domain and range.
Let's look at each function separately:
1. f(x) = x^2 - 1 ; x = -3, -2, -1, 0, 1, 2, 3
- Calculate the values of y for each given value of x:
f(-3) = (-3)^2 - 1 = 9 - 1 = 8
f(-2) = (-2)^2 - 1 = 4 - 1 = 3
f(-1) = (-1)^2 - 1 = 1 - 1 = 0
f(0) = (0)^2 - 1 = 0 - 1 = -1
f(1) = (1)^2 - 1 = 1 - 1 = 0
f(2) = (2)^2 - 1 = 4 - 1 = 3
f(3) = (3)^2 - 1 = 9 - 1 = 8
- Plot the points: (-3, 8), (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3), (3, 8)
- Connect the points to form a curve.
- The graph is a parabolic curve opening upwards.
- The domain is all real numbers (-∞, ∞), as there are no restrictions on the x-values.
- The range is [−1, ∞), as the curve extends indefinitely upwards but does not go below -1.
2. g(x) = √(x+1) - 2 ; x = -1, 0, 3, 8
- Calculate the values of y for each given value of x:
g(-1) = √((-1)+1) - 2 = √0 - 2 = -2
g(0) = √((0)+1) - 2 = √1 - 2 = -1
g(3) = √((3)+1) - 2 = √4 - 2 = 0
g(8) = √((8)+1) - 2 = √9 - 2 = 1
- Plot the points: (-1, -2), (0, -1), (3, 0), (8, 1)
- Connect the points to form a curve.
- The graph is a square root function shifted horizontally by 1 unit to the left and vertically by 2 units downwards.
- The domain is [−1, ∞), as the square root function is only defined for or equal to zero or positive values.
- The range is [-2, 1], as the vertical shift of -2 results in the graph never going above -2 and the maximum output value being 1.
3. h(x) = |x-3| ; x = 0, 1, 2, 3, 4, 5, 6
- Calculate the values of y for each given value of x:
h(0) = |0-3| = 3
h(1) = |1-3| = 2
h(2) = |2-3| = 1
h(3) = |3-3| = 0
h(4) = |4-3| = 1
h(5) = |5-3| = 2
h(6) = |6-3| = 3
- Plot the points: (0, 3), (1, 2), (2, 1), (3, 0), (4, 1), (5, 2), (6, 3)
- Connect the points to form a "V" shaped graph.
- The graph is a V-shaped graph symmetric to the vertical line x = 3.
- The domain is all real numbers (-∞, ∞), as there are no restrictions on the x-values.
- The range is [0, ∞), as the output values are always positive or zero.