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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

Best Answer 

 #2
avatar+128399 
+3

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

 

See an example  here

 

 

 

cool cool cool

 Mar 12, 2024
 #1
avatar+116 
+3

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
avatar+128399 
+3
Best Answer

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

 

See an example  here

 

 

 

cool cool cool

CPhill Mar 12, 2024
 #5
avatar+16 
-2

AoPS homework again, "NotLatePY"?

 Mar 15, 2024
edited by Holtran  Mar 15, 2024
 #7
avatar+16 
-2

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

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