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Magazine Chapter 3: Problem 22
Sketch the graph of the function by first making a table of values. $$ g(x)=\frac{|x|}{x^{2}} $$
Short Answer
Expert verified
The graph has asymptotes at \( x = 0 \); it rises vertically when \( x \to 0^+ \) and falls vertically when \( x \to 0^- \).
Step by step solution
01
Understand the Function
We need to sketch the graph of \( g(x) = \frac{|x|}{x^2} \). This function behaves differently depending on whether \( x \) is positive, negative, or zero. For positive \( x \), \( |x| = x \) and for negative \( x \), \( |x| = -x \). We cannot evaluate the function at \( x = 0 \) since the denominator would be zero.
02
Analyze the function for positive values of x
For \( x > 0 \), the function simplifies to \( g(x) = \frac{x}{x^2} = \frac{1}{x} \). Analyze its behavior as \( x \to 0^+ \) and as \( x \to \infty \): - \( x \to 0^+ \): \( g(x) \to \infty \)- \( x \to \infty \): \( g(x) \to 0 \)
03
Analyze the function for negative values of x
For \( x < 0 \), the function simplifies to \( g(x) = \frac{-x}{x^2} = -\frac{1}{x} \). Analyze its behavior as \( x \to 0^- \) and as \( x \to -\infty \): - \( x \to 0^- \): \( g(x) \to -\infty \)- \( x \to -\infty \): \( g(x) \to 0 \)
04
Creating a Table of Values
Select key points to evaluate the function: - \( x = -2, -1, -0.5, 0.5, 1, 2 \)Calculate:- \( x = -2 \): \( g(x) = -\frac{1}{2} \)- \( x = -1 \): \( g(x) = -1 \)- \( x = -0.5 \): \( g(x) = -2 \)- \( x = 0.5 \): \( g(x) = 2 \)- \( x = 1 \): \( g(x) = 1 \)- \( x = 2 \): \( g(x) = \frac{1}{2} \)
05
Sketching the Graph
Plot the points from the table of values on a graph. - As \( x \to 0^+ \), the curve approaches \( y \to \infty \).- As \( x \to 0^- \), the curve approaches \( y \to -\infty \).- The graph has no point at \( x = 0 \) since the function is undefined there. Verify these trends using the table values, and draw the curves towards these asymptotic behaviors.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value
In mathematics, the absolute value of a number is its distance from zero on a number line, disregarding its direction. For instance, the absolute value of both 5 and -5 is 5. In the function we are exploring, \(g(x) = \frac{|x|}{x^2}\), the absolute value alters how the numerator behaves.
- When \(x\) is positive, \(|x| = x\). This means for positive \(x\), the function simplifies to \(g(x) = \frac{x}{x^2} = \frac{1}{x}\).
- For negative values of \(x\), \(|x| = -x\), leading the function to simplify to \(g(x) = \frac{-x}{x^2} = -\frac{1}{x}\).
Understanding absolute value helps us examine the symmetry of the function around the y-axis, showing us how the positive and negative x-values will affect the graph of \(g(x)\). It ensures that we account for both positive and negative aspects, maintaining the function's proper shape.
- When \(x\) is positive, \(|x| = x\). This means for positive \(x\), the function simplifies to \(g(x) = \frac{x}{x^2} = \frac{1}{x}\).
- For negative values of \(x\), \(|x| = -x\), leading the function to simplify to \(g(x) = \frac{-x}{x^2} = -\frac{1}{x}\).
Understanding absolute value helps us examine the symmetry of the function around the y-axis, showing us how the positive and negative x-values will affect the graph of \(g(x)\). It ensures that we account for both positive and negative aspects, maintaining the function's proper shape.
Asymptotic Behavior
Asymptotic behavior refers to how a function behaves as it approaches a line or a point it never actually reaches. Understanding this behavior is crucial for graphing rational functions like \(g(x) = \frac{|x|}{x^2}\). The function showcases vertical asymptotic behavior as it approaches zero from both sides.
- As \(x\) approaches \(0^+\), \(g(x)\) tends toward positive infinity. This means the graph shoots upward vertically without bound, indicating a vertical asymptote at \(x = 0\).
- As \(x\) approaches \(0^-\), \(g(x)\) tends toward negative infinity. This reveals another vertical asymptote as the graph dives downward.
- As \(x\) becomes very large (positively or negatively), \(g(x)\) approaches zero. Thus, there is a horizontal asymptote at \(y = 0\).
Table of Values
Creating a table of values is a vital step for graphing any function, as it provides specific data points that inform the shape of the graph. By substituting specific \(x\) values into the function \(g(x) = \frac{|x|}{x^2}\), we calculate corresponding \(y\) values, which we then plot.
Here are the data points we can use for our graph:
Here are the data points we can use for our graph:
- When \(x = -2\), \(g(x) = -\frac{1}{2}\)
- When \(x = -1\), \(g(x) = -1\)
- When \(x = -0.5\), \(g(x) = -2\)
- When \(x = 0.5\), \(g(x) = 2\)
- When \(x = 1\), \(g(x) = 1\)
- When \(x = 2\), \(g(x) = \frac{1}{2}\)
Undefined Points
In the function \(g(x) = \frac{|x|}{x^2}\), an undefined point occurs when the function cannot be evaluated at a particular \(x\) value. Here, this happens when the denominator equals zero, making the function undefined at \(x = 0\) since division by zero is impossible.
- Specifically, \(g(x)\) becomes \(\frac{|0|}{0^2}\), leading to an undefined result.
This lack of definition at \(x = 0\) introduces a vertical asymptote. The graph approaches infinity from either side without actually touching, highlighting the point as undefined.
Understanding undefined points and their implications on a graph aids in avoiding errors while sketching and interpreting rational functions. Recognizing these points allows us to know where the function doesn't exist and ensures we manage these discontinuities properly.
- Specifically, \(g(x)\) becomes \(\frac{|0|}{0^2}\), leading to an undefined result.
This lack of definition at \(x = 0\) introduces a vertical asymptote. The graph approaches infinity from either side without actually touching, highlighting the point as undefined.
Understanding undefined points and their implications on a graph aids in avoiding errors while sketching and interpreting rational functions. Recognizing these points allows us to know where the function doesn't exist and ensures we manage these discontinuities properly.
