what is the domain of each of the following functions,their intercepts and thier range?
f(x)=x+3 and f(x)=-2x-3 if x<-2 and x>=-2
All polynomials have a domain of all reals numbers.
Just plug in x=0 to get the y-intercept, and f(x)=0 to get the x-intercept.
Since these are just lines, the range is also all real numbers.
Not sure what help the "if x<-2 and x>=-2" provides, since that pretty much includes all real numbers.
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To determine the domain, intercepts, and range of a function, we need to analyze the given equations:
1. f(x) = x + 3
Domain: The domain of a function refers to all possible values of x for which the function is defined. In this case, the equation is defined for all real numbers since there are no restrictions or undefined values.
Intercepts: To find the x-intercept, we set f(x) = 0 and solve for x.
0 = x + 3
x = -3
So, the x-intercept is at (-3, 0).
To find the y-intercept, we set x = 0 and solve for f(x).
f(0) = 0 + 3
f(0) = 3
So, the y-intercept is at (0, 3).
Range: The range of a function represents all possible values that f(x) can take. In this case, since the function f(x) = x + 3 is a linear function with a positive slope, it will continue infinitely upward. Therefore, the range is (-∞, +∞).
2. f(x) = -2x - 3, if x < -2
f(x) = x + 3, if x ≥ -2
Domain: For the first equation, f(x) = -2x - 3, if x < -2, the function is defined for all x-values less than -2. For the second equation, f(x) = x + 3, if x ≥ -2, the function is defined for all x-values greater than or equal to -2. Therefore, the domain is (-∞, -2) ∪ [-2, +∞).
Intercepts:
For f(x) = -2x - 3:
To find the x-intercept, we set f(x) = 0 and solve for x.
0 = -2x - 3
2x = -3
x = -3/2 or -1.5
So, the x-intercept is at (-1.5, 0).
To find the y-intercept, we set x = 0 and solve for f(x).
f(0) = -2(0) - 3
f(0) = -3
So, the y-intercept is at (0, -3).
For f(x) = x + 3:
The intercepts for this equation have already been found in the previous example, which are (-3, 0) for the x-intercept and (0, 3) for the y-intercept.
Range:
The first equation, f(x) = -2x - 3, if x < -2, represents a downward-sloping line. Therefore, the range is (-∞, -3].
The second equation, f(x) = x + 3, if x ≥ -2, represents an upward-sloping line. Therefore, the range is [-2, +∞).