To find the domain of a piecewise function like f(x), we need to determine the values of x for which the function is defined. In this case, we have three separate conditions for the function f(x), so we need to find the overlap of their domains.
1. For the first condition, f(x) = (x+10) is defined for x values less than -2 because the expression is valid for all values less than -2. So the domain for this part is (-β, -2).
2. For the second condition, f(x) = 3 is defined for x values between -2 and 0, including -2 but excluding 0. So the domain for this part is [-2, 0).
3. For the third condition, f(x) = (2x-6) is defined for x values between 0 and 5, including 0 but excluding 5. So the domain for this part is [0, 5).
By combining all the domains, the overall domain of f(x) is (-β, -2) βͺ [-2, 0) βͺ [0, 5).
To find the range of the function, we need to consider the outputs or values that f(x) can take. Since the function is defined separately for different intervals, let's consider each interval.
1. For f(x) = (x+10) when x < -2, the function takes all possible values from negative infinity to positive infinity.
2. For f(x) = 3 when -2 β€ x < 0, the function takes a constant value of 3.
3. For f(x) = (2x-6) when 0 < x β€ 5, the function takes all possible values from -6 to 4, inclusive.
Therefore, the range of the function is (-β, β) βͺ {3} βͺ [-6, 4].
To find the x-intercepts, we need to find the values of x where f(x) = 0. Looking at the function, we can see that there is no part of the function that equals zero. So, there are no x-intercepts.
To find the y-intercept, we need to find the value of f(x) when x = 0. Using the third condition of the function, f(0) = (2(0) - 6) = -6. Therefore, the y-intercept is (0, -6).
To determine if the function is continuous on its domain, we need to check if there are any jumps, breaks, or removable discontinuities in the function. In this case, the function is continuous for each interval since each part is defined by a continuous equation. So, f(x) is continuous on its domain.
To graph the function, plot the points and connect them with appropriate line segments based on the domain and range restrictions mentioned above. The graph will consist of three separate line segments: one with a slope of 1 from x = -9 to x = -2; a horizontal line at y = 3 from x = -2 to x = 0; and a line with a slope of 2 from x = 0 to x = 5.