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Chapter 4: Problem 39

Exer. 35-46: Sketch the graph of \(f\) $$f(x)=\log _{3}\left(x^{2}\right)$$

Short Answer

Expert verified
The graph of \( f(x) = \log_3(x^2) \) is even, passing through \((1,0)\) and \((-1,0)\), with asymptotes approaching \(-\infty\) near zero, symmetric about the y-axis.

Step by step solution

01

Understand the Function

The function given is \( f(x) = \log_{3}(x^2) \). This is a logarithmic function with base 3, which means it will be defined for \( x^2 > 0 \), or in simpler terms, when \( x eq 0 \).
02

Identify the Domain

Since \( x^2 > 0 \) for all \( x eq 0 \), the domain of \( f(x) = \log_{3}(x^2) \) includes all non-zero real numbers: \( x \in \mathbb{R} \setminus \{0\} \).
03

Behavior at Key Points

Calculate \( f(x) \) at key values of \( x \). For instance, at \( x=1 \), \( f(1) = \log_3(1^2) = 0 \), and similarly \( f(-1) = \log_3((-1)^2) = 0 \). This suggests \( x = 1 \) and \( x = -1 \) are points on the graph. Additionally, as \( x \) approaches 0, \( f(x) \) will tend towards \( -\infty \).
04

Symmetry of the Function

Observe that \( f(-x) = \log_3((-x)^2) = \log_3(x^2) = f(x) \). This indicates that the function is even, leading to symmetry about the y-axis.
05

Sketch the Graph

Using the information from previous steps, sketch the graph. It will pass through the points \((1,0)\) and \((-1,0)\), display symmetry about the y-axis, and tend toward \(-\infty\) as \( x \to 0^+ \) and \( x \to 0^- \). The graph will rise as \( |x| \) increases.

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Key Concepts

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

Graphing Logarithmic Functions
Graphing logarithmic functions, such as the one in our exercise, offers a unique insight into how logarithmic equations behave visually. The function of interest is \( f(x) = \log_{3}(x^2) \). This indicates that we're observing a logarithmic function with a base 3. Here's how you can sketch it:
  • Plot key points: It helps to identify and plot essential points, such as \( (1, 0) \) and \( (-1, 0) \), because they tell us where the graph intersects the x-axis.
  • Analyze behavior: Understand that as \( |x| \) becomes larger, \( f(x) \) increases upwards. This means the graph will continue growing as the absolute value of \( x \) moves away from zero.
  • Understand the asymptotic behavior: As \( x \) approaches zero, \( f(x) \) will tend towards \( -\infty \), which helps to visualize the graph sloping sharply downward on either side of the y-axis.
By considering these aspects, you can create a clean and accurate graph of the function. This is crucial for visualizing how the function behaves over its domain.
Domain of a Function
The domain of a function represents all possible input values (x-values) for which the function is defined. For the function \( f(x) = \log_{3}(x^2) \), determining the domain requires understanding the constraints placed by the logarithmic and squared components.
  • Consider the logarithm: Logarithms require positive arguments, so \( x^2 > 0 \).
  • Analyze the squared term: Since \( x^2 \) is positive for all values except when \( x = 0 \), the function is undefined at \( x = 0 \).
Therefore, the domain of \( f(x) = \log_{3}(x^2) \) is all real numbers except zero, written as \( x \in \mathbb{R} \setminus \{0\} \). Understanding the domain is essential for knowing where your function lives and breathes, guiding you on what to expect from the function's behavior across its allowable values.
Even Functions
Even functions possess a special property of symmetry around the y-axis. An even function satisfies the condition \( f(x) = f(-x) \) for all \( x \) in its domain. Upon inspecting the function \( f(x) = \log_{3}(x^2) \), you can see it is even for several reasons:
  • Function relation: Since \( f(-x) = \log_{3}((-x)^2) = \log_{3}(x^2) = f(x) \), it clearly satisfies the even function condition.
  • Graphical symmetry: The symmetry about the y-axis indicates that the graph of \( f(x) = \log_{3}(x^2) \) is a mirrored image when reflected over the y-axis.
Recognizing even functions is crucial for simplifying graph sketches and predictions about the function's behavior, further revealing the elegance of the underlying mathematical properties.
Symmetry in Graphs
Symmetry in graphs is a fascinating concept where certain transformations leave the graph unchanged. In our function, \( f(x) = \log_{3}(x^2) \), there's symmetry about the y-axis due to its even nature.
  • Identify symmetrical attributes: The even function property \( f(x) = f(-x) \) confirms this symmetry. Visually, this means the graph appears identical on both sides of the y-axis.
  • Graphical implications: This symmetry implies that for every point \( (a, b) \) on the graph, there exists a point \( (-a, b) \) too.
Understanding such symmetry can simplify both graphing tasks and analytical pursuits. It provides a clearer picture of different sections of the graph, empowering us to grasp the full span of the function's behavior effortlessly.

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Most popular questions from this chapter

Approximate the function at the value of \(x\) to four decimal places. (a) \(f(x)=\ln (x+1)+e^{x}, \quad x=2\) (b) \(g(x)=\frac{(\log x)^{2}-\log x}{4}, \quad x=3.97\)

Graph \(f\) and \(g\) on the same coordinate plane, and estimate the solutions of the equation \(f(x)=g(x)\). $$f(x)=0.3 e^{x} ; \quad g(x)=x^{3}-x$$

Graph \(f\) and \(g\) on the same coordinate plane, and estimate the solution of the inequality \(f(x) \geq g(x)\). $$f(x)=x \ln |x| ; \quad g(x)=0.15 e^{x}$$

Sketch the graph of \(f\). $$f(x)=\log (x+10)$$

Use the graph of \(f\) to determine whether \(f\) is one-to-one. $$f(x)=\frac{x-8}{x^{2 / 3}+4}$$
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