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# Domain Question

0
11
8
+46

If $$f(x)$$ is a function whose domain is $$[-8, 8]$$, and $$g(x)=f\left(\frac x2\right)$$, then the domain of $$g(x)$$ is an interval of what width?

Thanks in advance :D

Mar 12, 2024

#2
+129799
+4

The  " x/2 " will double the interval width of f(x).....so the interval width = 32

See an example  here

Mar 12, 2024

#1
+193
+4

The strategy is to consider how values are scaled when moving from the domain of f to the domain of g.

Since g(x)=f(x/2), taking an input value x in the domain of g and feeding it into f is the same as feeding x/2 into f.

The domain of f is [−8,8], which means the valid inputs for f range from −8 to 8. When we take these values and divide them by 2, we get a range of values from 2−8​=−4 to 28​=4.

Now, these scaled values cover the entire domain of f because f is defined across this range. So the new interval we obtain, [−4,4], has the same width as the original domain [-8, 8]. The width is ${-8} - {-4} = 4$.

However, the new interval is centered at x=0 instead of x=4. So the resulting domain of g is [−4,4]​.

The interval [-4,4] has width 8.

Mar 12, 2024
#2
+129799
+4

The  " x/2 " will double the interval width of f(x).....so the interval width = 32

See an example  here

CPhill Mar 12, 2024
#5
+28
-1

AoPS homework again, "NotLatePY"?

Mar 15, 2024
edited by Holtran  Mar 15, 2024
#7
+28
-1

Thank you for your insightful snark, NotLatePY! I wonder when you will start completing your homework honestly instead of running here to cheat!

Mar 16, 2024
edited by Holtran  Mar 16, 2024