Question

If ( f(x) = x^4 + 5 ), ( g(x) = x - 5 ), ( h(x) = √{x} ), then ( f ˚ g ˚ h (x) ) equals:

A. ( x^2 + 5 )
B. ( x^2 - 5 )
C. ( x^4 + 5 )
D. ( x^4 - 5 )

Answer 1

To solve f ∘ g ∘ h (x), calculate each function in the order of h, g, and f. First, find the square root of x, subtract 5, and then raise the result to the fourth power and add 5. The correct answer is x^2 - 5, where the constants cancel out.

Step-by-step explanation:

To find ( f ∘ g ∘ h (x) ) when given ( f(x) = x^4 + 5 ), ( g(x) = x - 5 ), and ( h(x) = √{x} ), we need to perform the function composition step by step.

First, we find g ∘ h (x):

1. h(x) is the square root of x, i.e., √{x}.
2. g(x) takes an input and subtracts 5, so g ∘ h (x) equates to (√{x}) - 5.

Next, we find f ∘ (g ∘ h (x)):

1. f(x) takes an input and applies the function x^4 + 5 to it.
2. Inputting g ∘ h (x) into f, we get f(g ∘ h (x)) = f((√{x}) - 5) = ((√{x}) - 5)^4 + 5.
3. Since (√{x})^2 = x, the equation simplifies to x^2 - 5 + 5.
4. With +5 and -5 cancelling each other out, we are left with x^2.

Therefore, the correct answer is B. ( x^2 - 5 ), with the understanding that the constant terms have cancelled each other out.