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Chapter 6: Problem 25

In Problems \(25-32\), find the domain of the given function \(f .\) Find the \(x\) -intercept and the vertical asymptote of the graph. Use transformations to graph the given function \(f\). $$ f(x)=-\log _{2} x $$

Short Answer

Expert verified
The function's domain is \( x > 0 \), the x-intercept is at (1,0), and there is a vertical asymptote at \( x = 0 \).

Step by step solution

01

Determine the Domain

The function given is \( f(x) = -\log_{2} x \). The domain of a logarithmic function \( \log_{b} x \) is \( x > 0 \). Therefore, the domain of \( f(x) = -\log_{2} x \) is \( x > 0 \).
02

Find the x-intercept

The \( x \)-intercept occurs where the function \( f(x) \) is equal to zero: \(-\log_{2} x = 0\). Solving for \( x \) gives \( \log_{2} x = 0 \). Therefore, \( x = 2^{0} = 1 \). Hence, the \( x \)-intercept is at (1, 0).
03

Identify the Vertical Asymptote

Vertical asymptotes for \( \log \) functions occur at the boundary of the domain. Since the domain is \( x > 0 \), the vertical asymptote is at \( x = 0 \).
04

Graph Using Transformations

The base function is \( \log_{2} x \). The graph of \( -\log_{2} x \) involves reflecting the graph of \( \log_{2} x \) across the x-axis. Start with the point (1,0), which remains fixed, and apply the reflection to points such as (2,1) to (2,-1) and (0.5,-1) to (0.5,1). The vertical asymptote at \( x=0 \) remains the same.

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Key Concepts

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

Domain of a Function
In mathematics, the domain of a function refers to the set of input values for which the function is defined. For logarithmic functions like \( f(x) = -\log_{2} x \), the domain is determined by the requirement that the argument inside the logarithm must be greater than zero. This means that for the function \( f(x) = -\log_{2} x \), the domain is \( x > 0 \). This restriction arises because you cannot take the logarithm of a non-positive number. Understanding the domain is crucial as it outlines the range of acceptable inputs for the function, ensuring that all calculations remain valid.
x-intercept
The x-intercept is the point where the graph of a function crosses the x-axis. To find the x-intercept, we need to set the function equal to zero and solve for \( x \). For the function \( f(x) = -\log_{2} x \), set \(-\log_{2} x = 0\). Solving this equation, you find \( \log_{2} x = 0 \), which implies that \( x = 2^0 = 1 \). Thus, the x-intercept of the function is at the point (1, 0). This point is essential when plotting the graph, as it provides a fixed reference where the graph touches the x-axis.
Vertical Asymptotes
Vertical asymptotes are values of \( x \) where the function becomes undefined or approaches infinity. For logarithmic functions, vertical asymptotes are found at the boundaries of the domain. For \( f(x) = -\log_{2} x \), since the domain is \( x > 0 \), the vertical asymptote is at \( x = 0 \). As \( x \) approaches this boundary, the function values tend to infinity in either direction. The presence of a vertical asymptote significantly influences the shape and direction of the graph, as the curve gets infinitely close to the line \( x = 0 \) but never touches it.
Graph Transformations
Graph transformations allow us to modify graphs in systematic ways. For \( f(x) = -\log_{2} x \), we start with the base graph of \( \log_{2} x \). This function is then transformed by reflecting it over the x-axis. Reflection involves flipping the graph upside down. Start with known points on the base graph, such as (1,0), (2,1), and (0.5,-1), and apply the reflection. The point (1,0) remains the same because it's on the axis of reflection, while (2,1) becomes (2,-1), and (0.5,-1) becomes (0.5,1). The vertical asymptote at \( x = 0 \) remains unchanged. These transformations help in understanding how different operations affect the shape and position of a graph.

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Most popular questions from this chapter

Solve the given logarithmic equation. $$ \log _{8} x+\log _{8} x^{2}=1 $$

Either use factoring or the quadratic formula to solve the given equation. $$ \left(\log _{10} x\right)^{2}+\log _{10} x=2 $$

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Either use factoring or the quadratic formula to solve the given equation. $$ \left(5^{x}\right)^{2}-2\left(5^{x}\right)-1=0 $$

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