The function f is defined as follows:
f(x)={x+6 if -5 less than or equal to x<1
f(x)={9 if x=1
f(x)={-x+5 if x>1
a. find the domain
b. locate any intercepts, if any
domain: -5 <= x
intercepts: (-6,0),(5,0),(0,6)
To find the domain of a function, we need to determine the set of all possible values for x. In this case, we have three different pieces of the function with different rules for each interval. Let's consider each interval separately:
a) For -5 ≤ x < 1, the function is defined as f(x) = x + 6. In this interval, x can take any value between -5 and 1, including the endpoints. Therefore, the domain in this interval is -5 ≤ x < 1.
b) For x = 1, the function is defined as f(x) = 9. So the function is defined only for x = 1. Therefore, the domain in this case is x = 1.
c) For x > 1, the function is defined as f(x) = -x + 5. In this interval, x can take any value greater than 1. Therefore, the domain in this interval is x > 1.
Combining the domains from each interval, the overall domain of the function f(x) is given by -5 ≤ x < 1 or x = 1 or x > 1.
To find the intercepts, we need to look for the values of x where the function intersects the x-axis (y = 0) or the y-axis (x = 0).
a) For -5 ≤ x < 1, the function f(x) = x + 6. The x-intercept occurs when f(x) = 0, so we solve the equation x + 6 = 0:
x = -6. Therefore, there is an x-intercept at x = -6.
b) For x = 1, the function f(x) = 9. In this case, the function is a constant, which means it is a horizontal line parallel to the x-axis. Since the y-axis intercept occurs at x = 0, there is no x-intercept in this case.
c) For x > 1, the function f(x) = -x + 5. The x-intercept occurs when f(x) = 0, so we solve the equation -x + 5 = 0:
x = 5. Therefore, there is an x-intercept at x = 5.
To summarize, the function f(x) has an x-intercept at x = -6 and x = 5. There are no y-intercepts since f(x) is not dependent on the y-coordinate.