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Chapter 1: Problem 36

In Exercises \(21-38\), evaluate the functions for the specified values, if possible. $$f(x)=x^{2}+10 \quad g(x)=\sqrt{x-1}$$ $$g(f(\sqrt{7}))$$

Short Answer

Expert verified
The answer is 4.

Step by step solution

01

Evaluate the inner function

First, evaluate the function \( f \) at \( \sqrt{7} \). Since \( f(x) = x^2 + 10 \), we have:\[ f(\sqrt{7}) = (\sqrt{7})^2 + 10 = 7 + 10 = 17 \].
02

Evaluate the outer function

Now, use the result from Step 1 to evaluate the function \( g \). We need to find \( g(17) \), and given \( g(x) = \sqrt{x - 1} \), we have:\[ g(17) = \sqrt{17 - 1} = \sqrt{16} = 4 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Composition of Functions
The composition of functions involves combining two functions where the output of one function becomes the input for another. When you see an expression like \( g(f(x)) \), you're looking at a composition, which can also be interpreted as \( g \) of \( f \) of \( x \). This might sound complex, but the concept is easier to understand with a step-by-step approach.

To perform function composition:
  • Identify the inner function and the outer function. In this example, \( f(x) = x^2 + 10 \) is the inner function, and \( g(x) = \sqrt{x-1} \) is the outer function.
  • Begin with the inner function first. Compute \( f(x) \) for the given \( x \), then take the result and input it into the outer function \( g \).

This technique is essential in mathematics because it allows you to build complex functions from simpler ones, providing powerful analytical tools to solve diverse problems.
Inner Function Evaluation
Evaluating the inner function is the first step in function composition. The inner function's role is to take an initial input, process it, and then provide a new value that the outer function can use. In our example, the inner function \( f(x) = x^2 + 10 \) is evaluated at \( \sqrt{7} \).

Here's a simple breakdown of evaluating \( f(\sqrt{7}) \):
  • The expression \((\sqrt{7})^2\) simplifies to \(7\), since squaring a square root cancels the square root.
  • The equation then simplifies to \(7 + 10 = 17\).

This evaluation process transforms the initial input \( \sqrt{7} \) into \( 17 \), which then feeds into the outer function as its new input.
Outer Function Evaluation
After evaluating the inner function, the next step in the function composition is to handle the outer function. The outer function builds upon the result from the inner function. In our example, this is where we evaluate \( g(x) = \sqrt{x - 1} \) using the result from the inner function.

Let's look at evaluating \( g(17) \):
  • Substitute \( 17 \) (the output of the inner function) into the outer function \( g \).
  • Perform the calculation: \( g(17) = \sqrt{17 - 1} = \sqrt{16} \).
  • \( \sqrt{16} \) simplifies neatly to \( 4 \).

The outer function refines the output of the inner function into a further simplified result, completing the composition process.

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Most popular questions from this chapter

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