/*! 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 31 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 31

Sketch the graph of the function, given that \(f,\) \(\mathrm{F}\) and \(\mathrm{g}\) are defined as follows. (Hint: Start with the basic graphs in Figure 7 on page 149.) $$f(x)=|x| \quad F(x)=1 / x \quad g(x)=\sqrt{1-x^{2}}$$ $$y=F(x+3)$$

Short Answer

Expert verified
Shift the graph of the reciprocal function \(F(x)\) left by 3 units.

Step by step solution

01

Understand Basic Functions

The provided functions are basic mathematical functions. The function \(f(x) = |x|\) represents an absolute value function, \(F(x) = \frac{1}{x}\) is a reciprocal function, and \(g(x) = \sqrt{1-x^2}\) is a semi-circle function restricted to \([-1,1]\).
02

Analyze Transformation

The function \(y = F(x + 3)\) is a transformation of \(F(x)\). Adding 3 inside the function \(F(x) = \frac{1}{x}\) shifts the graph 3 units to the left. This means every point \((x,y)\) on the graph of \(F(x)\) is shifted to \((x-3, y)\).
03

Sketch the Basic Graph of \(F(x)\)

The graph of \(F(x) = \frac{1}{x}\) has two branches, one in the first quadrant and one in the third quadrant, with vertical asymptote at \(x = 0\) and horizontal asymptote at \(y = 0\). The graph approaches but never touches these asymptotes.
04

Apply the Transformation

To sketch \(y = F(x+3)\), shift the graph of \(F(x)\) 3 units left. The vertical asymptote moves from \(x = 0\) to \(x = -3\). The horizontal asymptote remains at \(y = 0\). This changes the position of the original graph to match the new function \(y = \frac{1}{x+3}\).
05

Finalize the Graph

Draw the final graph of \(y = \frac{1}{x+3}\). Position the branches of the reciprocal function according to the transformation shift. The function remains undefined at \(x = -3\) due to the vertical asymptote. Ensure the graph reflects the typical shape of a reciprocal function but shifted to the correct location.

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.

absolute value function
The absolute value function, given by \(f(x) = |x|\), is a fundamental concept in mathematics. It defines the distance of a number from zero without considering direction. This means that whether \(x\) is positive or negative, \(f(x)\) will always be non-negative.

Visualizing the absolute value function, its graph takes the form of a V-shape. This V-shape is due to the symmetrical effect the absolute value has on positive and negative inputs. To sketch this graph:
  • Draw two linear lines that intersect at the origin \((0, 0)\).
  • Both lines have a slope (or inclination) of 1 and -1 to the right and left of the origin, respectively.
  • The graph will continue upwards with stationary slopes in both directions.
Understanding the basic shape of this function is essential, as more complex functions can utilize these outlines to show behavior changes or transformations.
reciprocal function
The reciprocal function, expressed as \(F(x) = \frac{1}{x}\), is another key mathematical concept. Reciprocal functions have unique characteristics defined by their two branches and asymptotic behavior.

When dealing with \(F(x)\), note that:
  • The function is undefined at \(x = 0\), resulting in a vertical asymptote.
  • The graph will never touch the x-axis, creating a horizontal asymptote at \(y = 0\).
  • The graph has branches in the first and third quadrants of the coordinate plane, as a result of the function's nature.
For transformations, like in \(y = F(x+3)\), understanding how the function changes when shifted is crucial:
  • The expression \(x+3\) indicates a horizontal shift left by 3 units.
  • Vertical and horizontal asymptotes guide positioning; in this case, change the vertical position to \(x = -3\).
This understanding allows easier transformations for more complex functional behavior.
graph sketching
Graph sketching involves drawing the overall behavior of a function based on its characteristics and transformations. This skill is pivotal for understanding function behaviors visually.

When sketching any function, including transformed functions:
  • Start with the basic form of the function to know where to begin.
  • Identify any transformations such as shifts, stretches, or skews. Pay attention if they affect x-positions or y-positions.
  • Apply these transformations carefully 鈥� be it shifting, rotating, or reflecting 鈥� to the base graph.
  • Draw major features such as asymptotes for clear illustration paths.
For the specific example of \(y = F(x+3)\), sketch the base reciprocal graph first. Then, implement a left shift of 3 units. The vertical asymptote originally at \(x = 0\) is adjusted to \(x = -3\). The curve itself stays recognizable but holds a new spatial anchor. This process aids in precision while maintaining recognizable function patterns.

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 \(q(x)=a x^{2}+b x+c .\) Evaluate $$q\left(\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}\right)$$

By definition, a fixed point for the function \(f\) is a number \(x_{0}\) such that \(f\left(x_{0}\right)=x_{0} .\) For instance, to find any fixed points for the function \(f(x)=3 x-2,\) we write \(3 x_{0}-2=x_{0} .\) On solving this last equation, we find that \(x_{0}=1 .\) Thus, 1 is a fixed point for \(f .\) Calculate the fixed points (if any) for each function. (a) \(f(x)=6 x+10\) (b) \(g(x)=x^{2}-2 x-4\) (c) \(S(t)=t^{2}\) (d) \(R(z)=(z+1) /(z-1)\)

Let \(g(x)=4 x-1 .\) Find \(f(x),\) given that the equation \((g \circ f)(x)=x+5\) is true for all values of \(x\).

Sketch the graph of the function, given that \(f,\) \(\mathrm{F}\) and \(\mathrm{g}\) are defined as follows. (Hint: Start with the basic graphs in Figure 7 on page 149.) $$f(x)=|x| \quad F(x)=1 / x \quad g(x)=\sqrt{1-x^{2}}$$ $$y=1-g(x-2)$$

Use the given function and compute the first six iterates of each initial input \(x_{0}\). In cases in which a calculator answer contains four or more decimal places, round the final answer to three decimal places. (However, during the calculations, work with all of the decimal places that your calculator affords.) \(g(x)=\frac{1}{4} x+3\) (a) \(x_{0}=3\) (b) \(x_{0}=4\) (c) \(x_{0}=5\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

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.