/*! 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 32 A function \(f\) is given. (a)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 32

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f\) (b) Find the domain and range of \(f\) from the graph. $$f(x)=x^{2}+4$$

Short Answer

Expert verified
Domain: \((-\infty, \infty)\), Range: \([4, \infty)\).

Step by step solution

01

Understand the Function

The function given is \( f(x) = x^2 + 4 \). This is a quadratic function which is a simple parabola that opens upwards.
02

Graph the Function

Use a graphing calculator to plot the graph of the function \( f(x) = x^2 + 4 \). The vertex of the parabola is at (0, 4). The parabola opens upwards with its minimum point at the vertex.
03

Find the Domain

For a quadratic function of the form \( f(x) = x^2 + c \), the domain is all real numbers because you can input any real number for \( x \) and still get a valid output. Hence, the domain is \( (-fty, fty) \).
04

Find the Range

From the graph, observe that the minimum value of \( f(x) \) occurs at the vertex, which is \( c = 4 \). Since the parabola opens upwards, all values of \( f(x) \) are greater than or equal to 4. Thus, the range is \( [4, fty) \).

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.

graphing functions
Graphing functions is a fundamental skill in mathematics that transforms equations into visual representations. For the given function, \( f(x) = x^2 + 4 \), graphing helps to see the shape and behavior of the function. This specific function is a quadratic, so it forms a parabola.

To graph any function effectively, especially quadratic ones like \( f(x) = ax^2 + bx + c \), follow these steps:
  • Identify the Key Features: For quadratic functions, focus on the vertex, axis of symmetry, and direction of opening (up or down).
  • Plot the Vertex: The vertex of \( f(x) = x^2 + 4 \) is at (0, 4). This is where the parabola reaches its minimum point.
  • Determine the Direction: If the coefficient of \( x^2 \) is positive, the parabola opens upwards, as in our function.
  • Sketch the Curve: Use a graphing calculator for a precise plot, but remember the basic shape and direction.
Visualizing these elements enables a deeper understanding of the equation and its practical applications in examining trends or solving problems.
domain and range
Understanding the domain and range of a function is crucial for evaluating where the function operates and the outputs it can produce. The domain refers to all possible input values \( x \), while the range refers to all potential output values \( f(x) \). For the given quadratic function \( f(x) = x^2 + 4 \):
  • Domain: Quadratic functions like this are defined for all real numbers. Thus, the domain of \( f(x) = x^2 + 4 \) is \( (-\infty, \infty) \), indicating that you can substitute any real number for \( x \). There are no restrictions like square roots or fractions.
  • Range: Since the parabola opens upwards and its vertex is at \( y = 4 \), all output values \( f(x) \) will be 4 or greater. Therefore, the range is \( [4, \infty) \), highlighting that the minimum output the function can produce is 4, after which it increases infinitely.
A clear understanding of the domain and range not only helps in graphing but also in understanding the real-life implications of mathematical functions.
parabolas
Parabolas are U-shaped graphs that represent quadratic functions. They play a notable role in mathematics due to their symmetrical properties and appearance in various natural phenomena. Our function \( f(x) = x^2 + 4 \) forms a basic parabola.
  • Characteristics of Parabolas: They have a line of symmetry, which passes through the vertex. In \( f(x) = x^2 + 4 \), the line of symmetry is x=0 since the vertex is (0, 4).
  • The Vertex: This point is crucial as it represents the minimum (or maximum) value of the function. For this function, the vertex is a minimum at (0, 4).
  • Opening Direction: Parabolas can open upwards or downwards. If the \( x^2 \) term is positive, like in our function, the parabola opens upwards.
  • Applications: Parabolas find application in physics, engineering, and optics, such as the paths of projectiles or light reflection in car headlights.
Understanding these properties and their graphical representation is invaluable for solving mathematical problems and applying math concepts to real-world scenarios.

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

Marcello's Pizza charges a base price of \(\$ 7\) for a large pizza, plus \(\$ 2\) for each topping. Thus, if you order a large pizza with \(x\) toppings, the price of your pizza is given by the function \(f(x)=7+2 x .\) Find \(f^{-1} .\) What does the function \(f^{-1}\) represent?

The function \(I(x)=x\) is called the identity function. Show that for any function \(f\) we have \(f \circ I=f, I \circ f=f,\) and \(f \circ f^{-1}=f^{-1} \circ f=I .\) (This means that the identity function \(I\) behaves for functions and composition just like the number 1 behaves for real numbers and multiplication.)

Revenue, Cost, and Profit A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x<10,000\) ), then the price per sticker is \(0.15-0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x-0.0000005 x^{2}\) dollars. Use the fact that profit \(=\) revenue \(-\) cost to express \(P(x),\) the profit on an order of \(x\) stickers, as a difference of two functions of \(x .\)

Suppose that $$\begin{array}{l}g(x)=2 x+1 \\\h(x)=4 x^{2}+4 x+7\end{array}$$ Find a function \(f\) such that \(f \circ g=h .\) (Think about what operations you would have to perform on the formula for \(g\) to end up with the formula for \(h .\) ) Now suppose that $$\begin{array}{l}f(x)=3 x+5 \\\h(x)=3 x^{2}+3 x+2\end{array}$$ Use the same sort of reasoning to find a function \(g\) such that \(f \circ g=h\)

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$f(x)=x^{2}, \quad g(x)=\sqrt{x-3}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.