+77 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
+195 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.
