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Sagot :
Sure, let's tackle this step-by-step for each drop-down menu.
1. Value of [tex]\( f(g(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex]:
- First, we need to compute [tex]\( g(x) \)[/tex]: [tex]\( g(x) = \frac{1}{4} \sqrt{x} \)[/tex].
- Then, we plug [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]: [tex]\( f(g(x)) = f\left(\frac{1}{4} \sqrt{x}\right) \)[/tex].
- The function [tex]\( f(x) \)[/tex] is given by [tex]\( f(x) = 16 x^2 \)[/tex].
- Substituting [tex]\( \frac{1}{4} \sqrt{x} \)[/tex] into [tex]\( f(x) \)[/tex], we get [tex]\( f\left(\frac{1}{4} \sqrt{x}\right) = 16 \left(\frac{1}{4} \sqrt{x}\right)^2 = 16 \left(\frac{1}{16} x\right) = x/4 \)[/tex].
Therefore, the value of [tex]\( f(g(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex] is [tex]\( \boxed{\frac{x}{4}} \)[/tex].
2. Value of [tex]\( g(f(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex]:
- First, we need to compute [tex]\( f(x) \)[/tex]: [tex]\( f(x) = 16 x^2 \)[/tex].
- Then, we plug [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]: [tex]\( g(f(x)) = g(16 x^2) \)[/tex].
- The function [tex]\( g(x) \)[/tex] is given by [tex]\( g(x) = \frac{1}{4} \sqrt{x} \)[/tex].
- Substituting [tex]\( 16 x^2 \)[/tex] into [tex]\( g(x) \)[/tex], we get [tex]\( g(16 x^2) = \frac{1}{4} \sqrt{16 x^2} = \frac{1}{4} \cdot 4 x = x \)[/tex].
Therefore, the value of [tex]\( g(f(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex] is [tex]\( \boxed{x} \)[/tex].
3. Checking if [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverse functions:
- We need to verify if both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] hold for all [tex]\( x \geq 0 \)[/tex].
- From our calculations:
- [tex]\( f(g(x)) = x/4 \)[/tex], which is not equal to [tex]\( x \)[/tex] for all [tex]\( x \geq 0 \)[/tex].
- [tex]\( g(f(x)) = x \)[/tex], which is equal to [tex]\( x \)[/tex] for all [tex]\( x \geq 0 \)[/tex].
- Since [tex]\( f(g(x)) \)[/tex] does not equal [tex]\( x \)[/tex] for all [tex]\( x \geq 0 \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverse functions.
Therefore, functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are [tex]\( \boxed{\text{not}} \)[/tex] inverse functions.
So, summarizing the answers for the drop-down menus:
- For [tex]\( x \geq 0 \)[/tex], the value of [tex]\( f(g(x)) \)[/tex] is [tex]\( \frac{x}{4} \)[/tex].
- For [tex]\( x \geq 0 \)[/tex], the value of [tex]\( g(f(x)) \)[/tex] is [tex]\( x \)[/tex].
- For [tex]\( x \geq 0 \)[/tex], functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverse functions.
1. Value of [tex]\( f(g(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex]:
- First, we need to compute [tex]\( g(x) \)[/tex]: [tex]\( g(x) = \frac{1}{4} \sqrt{x} \)[/tex].
- Then, we plug [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]: [tex]\( f(g(x)) = f\left(\frac{1}{4} \sqrt{x}\right) \)[/tex].
- The function [tex]\( f(x) \)[/tex] is given by [tex]\( f(x) = 16 x^2 \)[/tex].
- Substituting [tex]\( \frac{1}{4} \sqrt{x} \)[/tex] into [tex]\( f(x) \)[/tex], we get [tex]\( f\left(\frac{1}{4} \sqrt{x}\right) = 16 \left(\frac{1}{4} \sqrt{x}\right)^2 = 16 \left(\frac{1}{16} x\right) = x/4 \)[/tex].
Therefore, the value of [tex]\( f(g(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex] is [tex]\( \boxed{\frac{x}{4}} \)[/tex].
2. Value of [tex]\( g(f(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex]:
- First, we need to compute [tex]\( f(x) \)[/tex]: [tex]\( f(x) = 16 x^2 \)[/tex].
- Then, we plug [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]: [tex]\( g(f(x)) = g(16 x^2) \)[/tex].
- The function [tex]\( g(x) \)[/tex] is given by [tex]\( g(x) = \frac{1}{4} \sqrt{x} \)[/tex].
- Substituting [tex]\( 16 x^2 \)[/tex] into [tex]\( g(x) \)[/tex], we get [tex]\( g(16 x^2) = \frac{1}{4} \sqrt{16 x^2} = \frac{1}{4} \cdot 4 x = x \)[/tex].
Therefore, the value of [tex]\( g(f(x)) \)[/tex] for [tex]\( x \geq 0 \)[/tex] is [tex]\( \boxed{x} \)[/tex].
3. Checking if [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverse functions:
- We need to verify if both [tex]\( f(g(x)) = x \)[/tex] and [tex]\( g(f(x)) = x \)[/tex] hold for all [tex]\( x \geq 0 \)[/tex].
- From our calculations:
- [tex]\( f(g(x)) = x/4 \)[/tex], which is not equal to [tex]\( x \)[/tex] for all [tex]\( x \geq 0 \)[/tex].
- [tex]\( g(f(x)) = x \)[/tex], which is equal to [tex]\( x \)[/tex] for all [tex]\( x \geq 0 \)[/tex].
- Since [tex]\( f(g(x)) \)[/tex] does not equal [tex]\( x \)[/tex] for all [tex]\( x \geq 0 \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverse functions.
Therefore, functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are [tex]\( \boxed{\text{not}} \)[/tex] inverse functions.
So, summarizing the answers for the drop-down menus:
- For [tex]\( x \geq 0 \)[/tex], the value of [tex]\( f(g(x)) \)[/tex] is [tex]\( \frac{x}{4} \)[/tex].
- For [tex]\( x \geq 0 \)[/tex], the value of [tex]\( g(f(x)) \)[/tex] is [tex]\( x \)[/tex].
- For [tex]\( x \geq 0 \)[/tex], functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not inverse functions.
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