/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 In Exercises \(21-38\), evaluate... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 27

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

Short Answer

Expert verified
\(\left(\frac{f}{g}\right)(10) = \frac{110}{3}\).

Step by step solution

01

Evaluate f(10)

First, we need to evaluate the function \( f(x) = x^2 + 10 \) at \( x = 10 \). Substitute 10 into the function: \[ f(10) = 10^2 + 10 = 100 + 10 = 110 \]
02

Evaluate g(10)

Next, we evaluate the function \( g(x) = \sqrt{x-1} \) at \( x = 10 \). Substitute 10 into the function: \[ g(10) = \sqrt{10 - 1} = \sqrt{9} = 3 \]
03

Compute \(\left(\frac{f}{g}\right)(10)\)

Now that we have both \( f(10) \) and \( g(10) \), we can find \( \left(\frac{f}{g}\right)(10) \) by dividing the two results: \[ \left(\frac{f}{g}\right)(10) = \frac{f(10)}{g(10)} = \frac{110}{3} \]
04

Simplify the final expression

The final step is to simplify the fraction \( \frac{110}{3} \) if needed. As 110 and 3 have no common factors, this fraction is already in simplest form, so:\[ \left(\frac{f}{g}\right)(10) = \frac{110}{3} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Function Evaluation
Function evaluation is the process of determining the output of a function for a particular input. This means substituting the input values into the function and calculating the result.
For example, let's take the function \( f(x) = x^2 + 10 \). If we want to evaluate \( f \) at \( x = 10 \), we simply substitute 10 in place of \( x \) and perform the calculation:
  • Substitute: \( f(10) = 10^2 + 10 \)
  • Calculate: \( 10^2 = 100 \)
  • Add: \( 100 + 10 = 110 \)
Similarly, for the function \( g(x) = \sqrt{x-1} \), substituting \( x = 10 \) yields \( g(10) = \sqrt{10 - 1} = \sqrt{9} = 3 \). This process of substitution and calculation is essential in determining individual function values.
Rational Functions
Rational functions are functions that can be expressed as the ratio of two polynomials. In the context of the exercise, we find the rational function \( \left(\frac{f}{g}\right)(x) \) by evaluating both \( f(x) \) and \( g(x) \) and then dividing the results.
To compute \( \left(\frac{f}{g}\right)(10) \):
  • First, evaluate \( f(x) = x^2 + 10 \) at \( x = 10 \) to find \( f(10) = 110 \).
  • Next, evaluate \( g(x) = \sqrt{x-1} \) at \( x = 10 \) to find \( g(10) = 3 \).
  • Then, divide these results: \( \left(\frac{f}{g}\right)(10) = \frac{110}{3} \).
The result \( \frac{110}{3} \) represents the output of the rational function for \( x = 10 \), simplified as needed.
Square Root Functions
Square root functions are particular types of functions that incorporate the square root operation. They are defined as \( g(x) = \sqrt{x-1} \) in this exercise. Understanding how to evaluate these functions can be important, as they have restrictions on their domain.
For example, \( g(x) \) is only defined when the expression under the square root is non-negative:
  • This means \( x - 1 \geq 0 \), or \( x \geq 1 \).
To evaluate \( g(10) \), knowing \( x = 10 \) satisfies this condition:
  • Substitute 10: \( g(10) = \sqrt{10 - 1} \).
  • Calculate: \( g(10) = \sqrt{9} = 3 \).
This shows square root functions can have less flexible domains than polynomial functions, making domain understanding crucial in problems involving them.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(f(x)=x^{2}+1 .\) Graph \(y_{1}=f(x)\) and \(y_{2}=f(x-2)\) in the same viewing window. Describe how the graph of \(y_{2}\) can be obtained from the graph of \(y_{1}\)

Refer to the following: In calculus, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of a function fis used to find the derivative \(f^{\prime}\) of \(f\), by allowing \(h\) to approach zero, \(h \rightarrow 0 .\) The derivative of the inverse function \(\left(f^{-1}\right)^{\prime}\) can be found using the formula $$\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$$ provided that the denominator is not 0 and both \(f\) and \(f^{-1}\) are differentiable. For the following one-to-one function, find (a) \(f^{-1},\) (b) \(f^{\prime},\) (c) \(\left(f^{-1}\right)^{\prime},\) and (d) verify the formula above. For (b) and (c), use the difference quotient. $$f(x)=\frac{1}{x+1}, x>-1$$

In calculus, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of a function \(f\) is used to find a new function \(f^{\prime},\) called the derivative off. To find \(f^{\prime},\) we let \(h\) approach \(0, h \rightarrow 0,\) in the difference quotient. For example, if \(f(x)=x^{2}, \frac{f(x+h)-f(x)}{h}=2 x+h,\) and allowing \(h=0,\) we have \(f^{\prime}(x)=2 x\) $$\text { Given } f(x)=\frac{x-5}{x+3}, \text { find } f^{\prime}(x)$$

Graph the functions \(f\) and \(g\) and the line \(y=x\) in the same screen. Do the two functions appear to be inverses of each other? $$f(x)=\sqrt[3]{x+3}-2 ; \quad g(x)=x^{3}+6 x^{2}+12 x+6$$

For Exercises \(111-114\), refer to the following: In calculus, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of a function \(f\) is used to find the derivative \(f^{\prime}\) of \(f\), by allowing \(h\) to approach zero, \(h \rightarrow 0 .\) Find the derivative of the following functions. \(f(x)=k,\) where \(k\) is a constant
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.