Math Problem Statement

like how do I do this question Find possible choices for outer and inner functions f and g such that the given function h equals

fcircle◦g.

h left parenthesis x right parenthesis equals StartRoot x Superscript 6 Baseline plus 7 EndRooth(x)=x6+7

Question content area bottom

Part 1

Choose the correct pair of functions below.

A.f left parenthesis x right parenthesis equals x cubed comma g left parenthesis x right parenthesis equals StartRoot x EndRoot minus 7

f left parenthesis x right parenthesis equals x cubed comma g left parenthesis x right parenthesis equals StartRoot x EndRoot minus 7f(x)=x3, g(x)=x−7

B.f left parenthesis x right parenthesis equals x Superscript 6 Baseline plus 7 comma g left parenthesis x right parenthesis equals StartRoot x EndRoot

f left parenthesis x right parenthesis equals x Superscript 6 Baseline plus 7 comma g left parenthesis x right parenthesis equals StartRoot x EndRootf(x)=x6+7, g(x)=x

C.f left parenthesis x right parenthesis equals StartRoot x EndRoot comma g left parenthesis x right parenthesis equals x Superscript 6 Baseline plus 7

f left parenthesis x right parenthesis equals StartRoot x EndRoot comma g left parenthesis x right parenthesis equals x Superscript 6 Baseline plus 7f(x)=x, g(x)=x6+7

D.f left parenthesis x right parenthesis equals StartRoot x EndRoot minus 7 comma g left parenthesis x right parenthesis equals x Superscript 6

f left parenthesis x right parenthesis equals StartRoot x EndRoot minus 7 comma g left parenthesis x right parenthesis equals x Superscript 6f(x)=x−7, g(x)=x6

Solution

The problem is asking you to find the correct combination of outer and inner functions f(x)f(x) and g(x)g(x) such that:

h(x)=x6+7h(x) = \sqrt{x^6 + 7} and we need to express h(x)h(x) as f(g(x))f(g(x)).

We can decompose the function step by step:

  1. Inner function g(x)g(x): First, notice that the expression inside the square root is x6+7x^6 + 7. So, it makes sense to choose g(x)=x6+7g(x) = x^6 + 7 as the inner function.

  2. Outer function f(x)f(x): The square root is applied to the entire inner expression. So, the outer function must be the square root function, f(x)=xf(x) = \sqrt{x}.

Thus, the function h(x)h(x) can be written as: h(x)=f(g(x))=x6+7h(x) = f(g(x)) = \sqrt{x^6 + 7}

Now, looking at the choices:

  • Choice B: f(x)=xf(x) = \sqrt{x} and g(x)=x6+7g(x) = x^6 + 7 is the correct pair.

Correct answer: B

Would you like more details on function composition or have any further questions? Here are 5 questions to consider next:

  1. How do you evaluate a composed function like f(g(x))f(g(x)) step-by-step?
  2. How do you determine the domain of a composed function?
  3. What are some practical uses of function composition in real-world problems?
  4. How can you verify that two functions are correctly composed?
  5. Can the composition of two functions result in a function that has a more restricted domain?

Tip: When dealing with function composition, always start by identifying the innermost operation and proceed outward.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Radicals
Polynomials

Formulas

f(g(x)) = √(x^6 + 7)
g(x) = x^6 + 7
f(x) = √x

Theorems

Composition of Functions
Radical Function Theorem

Suitable Grade Level

Grades 10-12

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