^{2}-7

^{2}) and g(x)= √(7-x)

**find a formula for h (x) and the domain of h**( use internal notation)

suppose that f(x) = √(x^{2} -7^{2}) and g(x)= √(7-x)

for each function (h) below,** find a formula for h (x) and the domain of h** ( use internal notation)

a) h(x) = f composed with g of (x)

b) h(x) = g composed with f of (x)

c) h(x)= f composed with f of (x)

d) h(x)= g composed with g of (x)

Tutors, sign in to answer this question.

Michael F. | Mathematics TutorMathematics Tutor

For f(x)=√(x²-7²) and g(x)= √(7-x)

a) h(x)=f(g(x))=√(g²(x)-7²)=√(7-x-7²)=√(7-x-49)=√(-x-42)

The domain are those x-values where -x-42≥0 or where x+42≥0 or where x≥-42

For these values h(x)=√(-x-42) takes values in the range [0,∞), or all positive values.

b) h(x)=g(f(x))=√(7-f(x))=√(7-√(x²-7²))=√(7-√(x²-49))

In order to determine the domain we must be able to take square roots so

1. x²-49 must be ≥0 or x²≥49 or either x≤-7 or x≥7 an in addition

√(x²-49) must be ≤7 since we need √(7-√(x²-49)).

√(x²-49)≤7 means that x²-49≤49 or that x²≤98 or that x≤√98=√(2×49)=7√2 or x≥-7√2

Combining the requirements the domain is the two intervals -7√2≤x≤-7 or 7≤x≤7√2

The range of h(x) is h(7) to h(7√2) or from 0 to √7. The function has the same range when the negative boundaries of the domain are used.

c) f(f(x))=√(f²(x)-49)=√(x²-49-49)=√(x²-98). The domain are those values where x²-98≥0 or where

x²≥98 or where x≥√98 or where x≤-√98, That is the two infinite intervals x≤-7√2 and x≥7√2

The range of the function is all the positive numbers.

d) g(g(x))=√(7-g(x))=√(7-√(7-x)). In order to evaluate the square roots we need to have x≤7 for √(7-x) and 7≥√(7-x) for the final square root. This last means that 49≥7-x or that x≥-42. The domain is

-42≤x≤7. The range is from 0 to √7

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.

Kevin S.

Personal tutoring: Series 3, 7, 63, 66, Accounting (CPA exam), Finance

$9.75 per 15 min

View Profile >

Alexis G.

All-Subject/Test Prep Tutor, Specializing in Language Instruction

$11.25 per 15 min

View Profile >

Cindy B.

Math, science, and English tutor; K-12 and remedial

$11.25 per 15 min

View Profile >