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Math  /  Algebra

QuestionFind functions ff and gg so that f∘g=Hf \circ g=H. H(x)=x2+17H(x)=\sqrt{x^{2}+17}
Choose the correct pair of functions. A. f(x)=x−17,g(x)=x2f(x)=\sqrt{x}-17, g(x)=x^{2} c. f(x)=x,g(x)=x2+17f(x)=\sqrt{x}, g(x)=x^{2}+17 B. f(x)=x2,g(x)=x−17f(x)=x^{2}, g(x)=\sqrt{x}-17 D. f(x)=x2+17,g(x)=xf(x)=x^{2}+17, g(x)=\sqrt{x}

Studdy Solution

STEP 1

1. We need to find functions f f and g g such that f∘g=H f \circ g = H .
2. The function H(x)=x2+17 H(x) = \sqrt{x^2 + 17} .
3. We are given multiple choices for f f and g g .

STEP 2

1. Understand the composition of functions.
2. Analyze each option to see if f(g(x))=H(x) f(g(x)) = H(x) .
3. Verify the correct pair of functions.

STEP 3

Understand that f∘g=H f \circ g = H means f(g(x))=H(x) f(g(x)) = H(x) .

STEP 4

Analyze Option A:
- f(x)=x−17 f(x) = \sqrt{x} - 17 - g(x)=x2 g(x) = x^2
Calculate f(g(x)) f(g(x)) :
f(g(x))=f(x2)=x2−17 f(g(x)) = f(x^2) = \sqrt{x^2} - 17
This simplifies to:
f(g(x))=∣x∣−17 f(g(x)) = |x| - 17
This does not equal H(x)=x2+17 H(x) = \sqrt{x^2 + 17} .

STEP 5

Analyze Option B:
- f(x)=x2 f(x) = x^2 - g(x)=x−17 g(x) = \sqrt{x} - 17
Calculate f(g(x)) f(g(x)) :
f(g(x))=f(x−17)=(x−17)2 f(g(x)) = f(\sqrt{x} - 17) = (\sqrt{x} - 17)^2
This does not equal H(x)=x2+17 H(x) = \sqrt{x^2 + 17} .

STEP 6

Analyze Option C:
- f(x)=x f(x) = \sqrt{x} - g(x)=x2+17 g(x) = x^2 + 17
Calculate f(g(x)) f(g(x)) :
f(g(x))=f(x2+17)=x2+17 f(g(x)) = f(x^2 + 17) = \sqrt{x^2 + 17}
This equals H(x)=x2+17 H(x) = \sqrt{x^2 + 17} .

STEP 7

Analyze Option D:
- f(x)=x2+17 f(x) = x^2 + 17 - g(x)=x g(x) = \sqrt{x}
Calculate f(g(x)) f(g(x)) :
f(g(x))=f(x)=(x)2+17=x+17 f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^2 + 17 = x + 17
This does not equal H(x)=x2+17 H(x) = \sqrt{x^2 + 17} .

STEP 8

Verify the correct pair of functions:
The correct pair of functions is Option C:
- f(x)=x f(x) = \sqrt{x} - g(x)=x2+17 g(x) = x^2 + 17
The correct pair of functions is:
Option C: f(x)=x, g(x)=x2+17 \boxed{\text{Option C: } f(x) = \sqrt{x}, \, g(x) = x^2 + 17}

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