given the following piecewise function
x+11 for -9<=x<-3
y= { 2 for -3<=x<=2
2x-6 for 2<x<=5
a. find the domain
b. find the range
c. find the intercepts
d. is f continuos on its domain. If not, state where f is discontinious
e. graph the function
This symbol { should be around all three lines.
PLEASE SHOW WORK
domain is all x where f is defined: [-9,5]
range: (-2,8]
intercepts: (3,0)
discontinuous at x=-3,2
a. To find the domain of the given piecewise function, we need to determine all the values of x for which the function is defined.
Looking at the conditions for each piece of the function:
1. For the first piece, x+11, it is defined for -9 ≤ x < -3.
2. For the second piece, 2, it is defined for -3 ≤ x ≤ 2.
3. For the third piece, 2x-6, it is defined for 2 < x ≤ 5.
Combining all the conditions, we get the domain as:
Domain: -9 ≤ x < -3 ∪ -3 ≤ x ≤ 2 ∪ 2 < x ≤ 5
b. To find the range of the function, we need to determine all the possible values of y for all the valid values of x in the domain.
1. For the first piece, y = x + 11, the range is all the possible values of y when -9 ≤ x < -3.
2. For the second piece, y = 2, the range is simply 2 as it doesn't depend on x.
3. For the third piece, y = 2x - 6, the range is all the possible values of y when 2 < x ≤ 5.
To find the range, we need to analyze the range for each piece. The range for each piece is as follows:
- For the first piece, the range is all real numbers such that y ≥ x + 11.
- For the second piece, the range is simply y = 2.
- For the third piece, the range is all real numbers such that y = 2x - 6.
Combining all the ranges, we get the overall range as:
Range: y ≥ -∞ ∪ y = 2 ∪ y = 2x - 6
c. To find the intercepts of the function, we need to determine the points where the graph of the function intersects the x-axis (x-intercepts) and the y-axis (y-intercepts).
1. X-intercepts: These occur when y = 0. To find the x-intercepts, we need to solve each piece separately:
First piece: x + 11 = 0
Solving, we get x = -11.
Second piece: There are no x-intercepts since y is always equal to 2.
Third piece: 2x - 6 = 0
Solving, we get x = 3.
Therefore, the x-intercepts are -11 and 3.
2. Y-intercept: This occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function:
First piece: y = 0 + 11 = 11.
Second piece: y = 2.
Third piece: y = 2(0) - 6 = -6.
Therefore, the y-intercept is 11 for the first piece.
d. To determine if the function is continuous on its domain, we need to check if there are any discontinuities or breaks in the function.
The function is continuous unless there are any gaps, jumps, or undefined points within the specified domain. In this case, it is continuous on its domain since there are no breakpoints or undefined points within the given conditions.
e. To graph the function, we can plot different parts of it based on the defined conditions:
- For the first piece, y = x + 11, graph the line for -9 ≤ x < -3.
- For the second piece, y = 2, graph a horizontal line for -3 ≤ x ≤ 2.
- For the third piece, y = 2x - 6, graph the line for 2 < x ≤ 5.
Combine all the different segments on the graph to represent the complete piecewise function according to the conditions given.