/*! 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 19 For each function: a. Evaluate... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 19

For each function: a. Evaluate the given expression. b. Find the domain of the function. c. Find the range. $$ f(x)=\sqrt{4-x^{2}} ; \text { find } f(0) $$

Short Answer

Expert verified
a) \( f(0) = 2 \); b) Domain: \([-2, 2]\); c) Range: \([0, 2]\).

Step by step solution

01

Evaluate the Expression at Given Point

To evaluate \( f(0) \), substitute \( x = 0 \) into the function: \( f(x) = \sqrt{4 - x^2} \). So, \( f(0) = \sqrt{4 - 0^2} = \sqrt{4} = 2 \).
02

Find the Domain of the Function

The expression inside the square root, \( 4 - x^2 \), must be greater than or equal to zero for \( f(x) \) to be defined. Hence, we set up the inequality: \( 4 - x^2 \geq 0 \). This simplifies to \( x^2 \leq 4 \), or \( -2 \leq x \leq 2 \). Thus, the domain of the function is \([-2, 2]\).
03

Find the Range of the Function

Since the expression inside the square root can vary between 0 and 4, we calculate the corresponding values of \( f(x) = \sqrt{4 - x^2} \). When \( x = 0 \), \( f(x) = 2 \); when \( x = \pm 2 \), \( f(x) = 0 \). Therefore, as \( x \) varies from \(-2\) to \(2\), \( f(x) \) varies from 0 to 2. Hence, the range of \( f(x) \) is \([0, 2]\).

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.

Domain of a Function
The domain of a function is the set of all possible input values (usually \( x \) values) that allow the function to work without any mathematical issues. For functions that include square roots, such as \( f(x) = \sqrt{4 - x^2} \), you need to make sure what's inside the square root stays non-negative.
For this function, we check when \( 4 - x^2 \geq 0 \) because square roots of negative numbers aren't real.
Solving \( 4 - x^2 \geq 0 \) gives us \( x^2 \leq 4 \), which further simplifies to \( -2 \leq x \leq 2 \).
Thus, the domain of this function is the set of \( x \) values from -2 to 2, inclusive.
  • Always ensure the inside of a square root is non-negative.
  • Domains keep functions defined and real.
Range of a Function
The range of a function is the set of all possible output values (\( y \) values) it can produce. Once we know the domain, we can find the range by examining the outputs based on the allowed inputs.
For \( f(x) = \sqrt{4 - x^2} \), the output values are influenced directly by what鈥檚 under the square root.
When \( x = 0 \), \( f(x) = 2 \), and when \( x = \pm 2 \), \( f(x) = 0 \). These values represent the extremes of the function as \( x \) varies from -2 to 2.
So, the function's range is \([0, 2]\).
  • Range represents all possible outputs given the domain.
  • Consider maximum and minimum values within the domain.
Evaluating Functions
Evaluating a function simply means finding the output for a given input. This involves substituting a specific input value into the function and simplifying.
Let鈥檚 evaluate \( f(0) \) in the function \( f(x) = \sqrt{4 - x^2} \).
  • Substitute \( x = 0 \) into the function: \( f(x) = \sqrt{4 - 0^2} = \sqrt{4} = 2 \).
  • This gives us the output \( f(0) = 2 \).
Understanding this helps in interpreting function behavior for individual points.
  • Substitute and calculate step-by-step.
  • Clear arithmetic helps avoid mistakes.

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

For each statement, state in words the values of \(x\) for which each exponential expression is defined. $$ x^{1 / 3} $$

91-92. BUSINESS: Phillips Curves Unemployment and inflation are inversely related, with one rising as the other falls, and an equation giving the relation is called a Phillips curve after the economist A. W. Phillips (1914-1975). Phillips used data from 1861 to 1957 to establish that in the United Kingdom the unemployment rate \(x\) and the wage inflation rate \(y\) were related by $$ y=9.638 x^{-1394}-0.900 $$ where \(x\) and \(y\) are both percents. Use this relation to estimate the inflation rate when the unemployment rate was a. 2 percent b. 5 percent

85-86.The intersection of an isocost line \(w L+r K=C\) and an isoquant curve \(K=a L^{b}\) (see pages 18 and 31 ) gives the amounts of labor \(L\) and capital \(K\) for fixed production and cost. Find the intersection point \((L, K)\) of each isocost and isoquant. [Hint: After substituting the second expression into the first, multiply through by \(L\) and factor.] $$ 3 L+8 K=48 \text { and } K=24 \cdot L^{-1} $$

Simplify. $$ \frac{\left(w w^{3}\right)^{2}}{w^{3} w^{2}} $$

GENERAL: Impact Velocity If a marble is dropped from a height of \(x\) feet, it will hit the ground with velocity \(v(x)=\frac{60}{11} \sqrt{x}\) miles per hour (neglecting air resistance). Use this formula to find the velocity with which a marble will strike the ground if it is dropped from the height of the tallest building in the United States, the 1776 -foot One World Trade Center in New York City.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.