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# Range and intercepts of a function- calculus

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How do you find the range and intercepts of the function f(x)=4|x+5| ?

Sep 17, 2017
edited by medlockb1234  Sep 17, 2017

#1
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The range is the set of all possible outputs that a function can produce.

$$f(x)=4|x+5|$$

The absolute value of any number will can result in a positive number or 0, and 4 multiplied by a positive number or 0 does not change this property at all. Therefore, the range is the following:

$$\text{Range}:\{\mathbb{R}|\hspace{1mm}y\geq0\}$$

This means that the range can be all nonnegative numbers. In interval notation, it would look like the following:

$$\text{Range}:[0,+\infty)$$

How do we find the intercepts? To do it without a graph, you can figure out by setting x=0 and y=0 and solving in each case. Let's do that.

 $$y=4|x+5|$$ Plug in 0 for x. This time, we are solving for the y-intercept. $$y=4|0+5|$$ Simplify inside the absolute value first. $$y=4|5|$$ $$y=4*5=20$$ We have now determined the coordinates of the y-intercept. $$(0,20)$$ This is the exact coordinates of the y-intercept.

Let's do the exact same process. This time, however, we set y=0 to find the x-intecept.

 $$y=4|x+5|$$ Set y equal to 0. $$0=4|x+5|$$ Divide by 4 on both sides. $$0=|x+5|$$ The absolute value always splits an equation into its plus or minus. However, 0 is neither positive nor negative, so there aren't 2 equations that one can set up. $$x+5=0$$ $$x=-5$$ We have now determined the x-intercept, as well. $$(-5,0)$$
Sep 17, 2017

#1
+2339
+2

The range is the set of all possible outputs that a function can produce.

$$f(x)=4|x+5|$$

The absolute value of any number will can result in a positive number or 0, and 4 multiplied by a positive number or 0 does not change this property at all. Therefore, the range is the following:

$$\text{Range}:\{\mathbb{R}|\hspace{1mm}y\geq0\}$$

This means that the range can be all nonnegative numbers. In interval notation, it would look like the following:

$$\text{Range}:[0,+\infty)$$

How do we find the intercepts? To do it without a graph, you can figure out by setting x=0 and y=0 and solving in each case. Let's do that.

 $$y=4|x+5|$$ Plug in 0 for x. This time, we are solving for the y-intercept. $$y=4|0+5|$$ Simplify inside the absolute value first. $$y=4|5|$$ $$y=4*5=20$$ We have now determined the coordinates of the y-intercept. $$(0,20)$$ This is the exact coordinates of the y-intercept.

Let's do the exact same process. This time, however, we set y=0 to find the x-intecept.

 $$y=4|x+5|$$ Set y equal to 0. $$0=4|x+5|$$ Divide by 4 on both sides. $$0=|x+5|$$ The absolute value always splits an equation into its plus or minus. However, 0 is neither positive nor negative, so there aren't 2 equations that one can set up. $$x+5=0$$ $$x=-5$$ We have now determined the x-intercept, as well. $$(-5,0)$$
TheXSquaredFactor Sep 17, 2017