/*! 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 23 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 23

Sketch the graph of the function using the approach presented in this section. $$f(x)=\frac{x}{(x+3)^{2}}$$

Short Answer

Expert verified
The graph of \(f(x)=\frac{x}{(x+3)^{2}}\) has a vertical asymptote at \(x = -3\), an x-intercept at \(x = 0\), and a y-intercept at \(y = 0\). The graph approaches but does not touch the vertical asymptote at \(x = -3\).

Step by step solution

01

Determine the Domain

The denominator of a fraction cannot be equal to zero because division by zero is undefined. Thus, to find the domain of \(f(x)=\frac{x}{(x+3)^{2}}\), set the denominator equal to zero and solve for x: \((x+3)^{2} = 0\). Solve this to find \(x= -3\). So, the domain of the function is \(x \in R, x 鈮� -3\).
02

Determine the Vertical Asymptote

A vertical asymptote occurs at the values of x where the function is undefined. From step 1, we know that x cannot be -3, so x = -3 is the vertical asymptote.
03

Determine the x-intercept

The x-intercept of the graph of a function is the value of x when the function is equal to 0. So, set the function equal to zero and solve for x: \(\frac{x}{(x+3)^{2}} = 0\). The solution is \(x=0\). Therefore, the x-intercept is at x=0.
04

Determine the y-intercept

The y-intercept of the graph of a function is the y value when x = 0. Substitute 0 into the function to get the y value: \(f(0) =\frac{0}{(0+3)^{2}} = 0\). So the y-intercept is at y = 0.
05

Sketch the graph

We can now sketch the graph using the information we have. The graph has a vertical asymptote at \(x = -3\), an x-intercept at \(x = 0\) and a y-intercept at \(y = 0\). The shape of the graph can be determined by looking at the general form of the function, which is a rational function. The graph will approach the vertical asymptote but never touch it. Draw the graph accordingly adjusting based on these features.

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Key Concepts

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

Domain of a Function
When finding the domain of a function, especially for rational functions, it is essential to spot any values that make the denominator zero. For the function \( f(x) = \frac{x}{(x+3)^2} \), identify the values of \(x\) that would result in division by zero since this is undefined in mathematics.
  • Set the denominator \((x+3)^2\) equal to zero.
  • Solve the equation to find \(x = -3\).
This means that at \( x = -3 \), the function is undefined. Thus, the domain of the function is all real numbers, except \(-3\). In set notation, this can be written as \( x \in \mathbb{R}, x eq -3 \). Remembering this step is crucial for graph sketching because it tells you where the graph cannot have any points.
Vertical Asymptote
Vertical asymptotes are lines that a graph approaches but never actually touches or crosses. They occur in rational functions at points where the function is undefined due to a zero in the denominator. For \( f(x) = \frac{x}{(x+3)^2} \), as identified in the domain, there is a vertical asymptote at \( x = -3 \).
  • As \(x\) gets closer to \(-3\), the value of the function grows infinitely large or small in magnitude.
  • The graph will approach, but never intersect this vertical line.
Visualizing a vertical asymptote helps in knowing the behavior of the graph around \( x = -3 \), ensuring a smoother and more accurate sketch.
X-Intercept
Finding the x-intercept involves setting the entire function equal to zero and solving for \(x\). With our function \( f(x) = \frac{x}{(x+3)^2} \), note that only the numerator can be zero for the whole fraction to equal zero.
  • Set \(x = 0\) because the numerator \(x\) must be zero.
  • So, the x-intercept is at the point \(x=0\).
This point on the graph is where the function crosses the x-axis and is a key feature for graph sketching, telling you exactly where the graph meets the horizontal line.
Y-Intercept
To find the y-intercept, substitute \( x = 0 \) into the function and simplify the expression. For \( f(x) = \frac{x}{(x+3)^2} \), when \( x = 0 \), it results in:
  • \( f(0) = \frac{0}{(0+3)^2} = 0 \).
  • Therefore, the y-intercept is at \( y = 0 \).
Identifying the y-intercept is key for graphing because it indicates where the curve crosses the y-axis. This intercept, together with the x-intercept, provides crucial points for accurately drawing the graph.
Rational Functions
Rational functions are ratios of two polynomials, written in the form \( \frac{f(x)}{g(x)} \). These functions, like \( f(x) = \frac{x}{(x+3)^2} \), are important as they can exhibit interesting behavior like vertical asymptotes, intercepts, and horizontal asymptotes.
  • The behavior of rational functions often involves creating vertical asymptotes where the denominator is zero.
  • These functions can have various shapes: simple lines, curves, or complex shapes depending on the degrees of the numerator and denominator.
Understanding rational functions lets you anticipate the general shape and behavior of the graph. It also aids in recognizing common features like asymptotes and intercepts, which helps significantly when sketching.

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Most popular questions from this chapter

Use a CAS to find the oblique asymptotes. Then use a graphing utility to draw the 2 graph of \(f\) and is asymptotes, and thereby confirm your findings. $$f(x)=\frac{3 x^{4}-4 x^{3}-2 x^{2}+2 x+2}{x^{3}-x}$$

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The lines \(y=(b / a) x\) and \(y=-(b / a) x\) are called asymtotes of the hyperbola $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $$ (a) Draw a figure that illustrates this asymptotic bchavior. (b) Show that the first-quadrant are of the hyperbola, the curve $$y=\frac{b}{a} \sqrt{x^{2}-a^{2}}$$ is indeed asymptotic to the line \(y=(b / a) x\) by showing that $$ \frac{b}{a} \sqrt{x^{2}-a^{2}}-\frac{b}{a} x \rightarrow 0 \text { as } x \rightarrow \infty $$ (c) Procceding as in part (b), show that the second-quadrant are of the hyperbola is asymplotic to the line \(y=\) \(-(b / a) x\) by taking a suitable limit as \(x \rightarrow-\infty\). (The asymptotic behavior in the other quadrants can be verified in an analogous manner, or by appealing to symmetry.)
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