/*! 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 12 For each function: a. Evaluate... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 12

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. [Hint: Use a graphing calculator.] $$ f(x)=\sqrt{x-4} ; \text { find } f(40) $$

Short Answer

Expert verified
f(40) = 6; Domain: [4, 鈭�); Range: [0, 鈭�)

Step by step solution

01

Evaluate the Expression

First, substitute the given value into the function.Substitute \( x = 40 \) into the function \( f(x) = \sqrt{x - 4} \). So, \( f(40) = \sqrt{40 - 4} = \sqrt{36} = 6 \).Therefore, \( f(40) = 6 \).
02

Determine the Domain

The domain of a function \( f(x) = \sqrt{x-4} \) consists of all \( x \) values that make the expression under the square root non-negative.That is, \( x - 4 \geq 0 \).Solving this inequality: \( x \geq 4 \).Thus, the domain is all real numbers \( x \) such that \( x \geq 4 \), represented as \([4, \infty)\).
03

Determine the Range Using Graph Analysis

The range of the function \( f(x) = \sqrt{x-4} \), for \( x \geq 4 \), starts at \( f(4) = \sqrt{4 - 4} = \sqrt{0} = 0 \) and can extend upwards to positive infinity.Since \( \sqrt{x-4} \) is always non-negative and increases as \( x \) increases, the range is \([0, \infty)\).Use a graphing calculator to confirm that the graph of \( f(x) \) supports this range.

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.

Evaluating Expressions
To evaluate an expression for a given function, we replace the variable with a specific value. Consider the function \( f(x) = \sqrt{x-4} \). When we need to find \( f(40) \), we substitute \( x = 40 \) into the function. This leads to the calculation \( f(40) = \sqrt{40 - 4} \).

Simplifying inside the square root gives us \( \sqrt{36} \), which results in 6, since 6 squared is 36. Thus, \( f(40) = 6 \). Evaluating expressions requires careful substitution and simplification to find the right answer.
Domain of a Function
The domain of a function includes all possible input values (\( x \)) that result in a real number output. For the function \( f(x) = \sqrt{x-4} \), we need to ensure that the expression inside the square root remains non-negative. Subtracting 4 from \( x \) should not result in a negative number, which means \( x - 4 \geq 0 \).

Solving this inequality, we find \( x \geq 4 \). Therefore, the domain in interval notation is \([4, \infty)\). When determining a domain, consider any restrictions the function might have, such as not allowing division by zero or taking the square root of a negative number.
Range of a Function
The range of a function is the set of all possible output values. For \( f(x) = \sqrt{x-4} \), as \( x \) starts at 4 and goes to infinity, the output of the function is always non-negative. Let's see why.

When \( x = 4 \), the function evaluates to 0, meaning \( f(4) = \sqrt{4-4} = 0 \). As \( x \) increases, \( \sqrt{x-4} \) becomes larger because you're taking the square root of greater numbers. This results in a range of \([0, \infty)\). Graphing the function can help confirm this range by visually showing that the function's outputs start at 0 and increase without bound as \( x \) grows.

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

How do the graphs of \(f(x)\) and \(f(x+10)+10\) differ?

Solve each equation by factoring. $$ 3 x^{5 / 2}-6 x^{3 / 2}=9 x^{1 / 2} $$

Find the slope (if it is defined) of the line determined by each pair of points. \(\left(-2, \frac{1}{2}\right)\) and \(\left(5, \frac{1}{2}\right)\)

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=\left\\{\begin{array}{ll} 3 x-1 & \text { if } x \geq 2 \\ 1-x & \text { if } x<2 \end{array}\right. $$

Write each interval in set notation and graph it on the real line. $$ (-3,5] $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.