The range of a function can be thought of as the set of -values given by the function's graph. For that reason, it's often helpful to plot the function when determining its range.
For example, consider the linear function
where is any real number. What is the range of ?
Since this is a linear function, we only need two points to plot it. For example:
When we have This gives the point
When we have This gives the point
Plotting these two points and connecting them, we obtain the following diagram.
Notice that the graph will cover all possible -values, so the range of the function is
Using interval notation, we can also write the range as
Let's now look at an example of a linear function with a restricted domain.
Consider the function for What is its range?
First, we graph the function.
Then, we find the smallest and largest -values on the graph.
The smallest -value is This occurs at the point However, the graph does not include this point.
The largest -value is This occurs at the point
The function covers all the -values between and including but not including
Therefore, the range is which can also be written as
Consider the linear function $f(x) = x-1,$ where $x$ is any real number. What is its range?
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$(-\infty, 1)$ |
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$ (-\infty, \infty)$ |
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$[1,\infty)$ |
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$[-1,\infty)$ |
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$(-\infty, -1]$ |
Consider the linear function $f(x) = \dfrac{1}{2}x+1,$ for $x \in (-2,2].$ What is its range?
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Consider the linear function $f(x) = 1 - \dfrac{1}{2}x,$ for $x \in (-2,2].$ What is its range?
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What is the range of the function shown below?
First, we find the smallest and largest -values in the graph.
- The smallest -value is This occurs when However, note that this point is not included in the graph.
- There is no largest -value.
The function covers all the -values between and but not including
Therefore, the range is which can also be written as
What is the range of the function $f(x),$ shown above?
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$0\lt f(x) \leq 4$ |
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$0 \leq f(x) \lt 4$ |
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$0 \lt f(x) \lt 4$ |
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$-\infty \lt f(x) \lt 4$ |
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$0 \leq f(x) \leq 4$ |
What is the range of the function $f(x),$ shown above?
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b
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What is the range of the function shown above?
We find the ranges of the individual pieces, and then calculate their union:
The range of the branch defined on is
The range of the branch defined on is
Therefore, the range of is
What is the range of the function $f(x),$ shown above?
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$[-2,3]$ |
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b
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$[-2,-1) \cup [1,3)$ |
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c
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$(-2,3)$ |
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d
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$[-2,3)$ |
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$[-2,-1] \cup [1,3]$ |
What is the range of the function $f(x),$ shown above?
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