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College Algebra

Section 9.3 Domain and Range of Transformations

Example 9.3.3.

Suppose we know that the domain of a function \(f(x)\) is \(\left(-2,5\right]\) and the range is \(\left[0,4\right)\text{.}\) We want to know the domain and range of the function \(g(x)=f(2x)+3\text{.}\)
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The first thing we need to do is figure out what tranformations took \(f\) to \(g\text{.}\) Using what you learned in the previous sections, we can see that we have:
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  • a vertical shift up by 3
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  • a horizontal compression by 2
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The vertical shift will move the whole graph up by 3, which means all of the y-values will increase by 3. So, the endpoints of the range will both increase by 3. Therefore, the range of \(g(x)\) is \(\left[3,7\right)\text{.}\)
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The horizontal compression will squish all of the graph towards the y-axis by a factor of 2, which means all of the x-values will be cut in half. So, the endpoints of the domain will both be cut in half. Therefore, the domain of \(g(x)\) is \(\left(-1,\frac{5}{2}\right]\text{.}\)
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You may find it helpful to draw a box to visualize the transformations. We don’t know exactly what the graph looks like, but because we know the domain and range, it is somewhere in the box below:
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After shifting it up by 3 and squishing towards the y-axis by 2, we end up with the following box, which gives us our new domain and range:
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Checkpoint 9.3.4.

Suppose the domain of a function \(f(x)\) is \(\left[3,7\right)\) and its range is \(\left[-1,3\right]\text{.}\) What is the domain and range of the function \(g(x)=-f(x+2)\text{?}\)
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Answer.
The domain of \(g(x)\) is \(\left[1,5\right)\) and its range is \(\left[-3,1\right]\text{.}\)
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Solution.
The first thing we need to do is figure out what tranformations took \(f\) to \(g\text{.}\) Using what you learned in the previous sections, we can see that we have:
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  • a vertical flip over the x-axis
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  • a horizontal shift left by 2
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The vertical flip will flip the entire graph over the x-axis, which means all of the y-values will change sign (negative becomes positive and positive becomes negative). So, the endpoints of the range will both change sign as well. Therefore, the range of \(g(x)\) is \(\left[-3,1\right]\text{.}\) Note that we still always write the lowest value first when writing interval notation.
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The horizontal shift will move the entire graph to the left by 2, which means all of the x-values will go down by 2. So, the endpoints of the domain will also decrease by 2. Therefore, the domain of \(g(x)\) is \(\left[1,5\right)\text{.}\)
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