Problem 73 - Precalculus with Limits

Question

True or False? In Exercises 73 and \(74,\) determinewhether the statement is true or false. Justify your answer. The line \(y=-2\) is an asymptote for the graph of \(f(x)=10^{x}-2\)

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understanding the Function and Asymptotes

The function given is \(f(x)=10^{x}-2\), which is an exponential function. An asymptote of a function is a line that the graph of the function approaches but never touches as it goes towards infinity or negative infinity. In this case, we need to determine whether \(y=-2\) is an asymptote for the graph of the given function.

Step 2: Identifying the Horizontal Asymptote

Exponential functions of the form \(y=ab^{x}-c\), where \(a\), \(b\), and \(c\) are constants, have a horizontal asymptote at \(y=-c\). This is because as \(x\) approaches infinity or negative infinity, the graph of the function approaches the value of \(c\), but never reaches it. Here, \(c\) is 2, so the horizontal asymptote of the function is at \(y=-2\). Therefore, the statement 'The line \(y=-2\) is an asymptote for the graph of \(f(x)=10^{x}-2\)' is true.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where the variable is in the exponent. They often appear in various real-world scenarios, such as population growth, radioactive decay, and compound interest. The general form of an exponential function is given by:

• \(f(x) = ab^x\)

where \(a\) represents the initial or starting value, \(b\) is the base representing the growth or decay factor, and \(x\) is the exponent or variable.
For the function \(f(x) = 10^x - 2\), it is written slightly differently, with \(-2\) acting as a vertical shift to the base function \(10^x\). Exponential functions have distinct characteristics:

• They either rapidly increase as \(x\) grows (if \(b > 1\)) or rapidly decrease (if \(0 < b < 1\)).
• The range of exponential functions is all real numbers above or below a horizontal asymptote.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches as \(x\) tends to positive or negative infinity. It essentially shows the behavior of the function at extreme values of \(x\).
For exponential functions of the form \(f(x) = ab^x - c\), the horizontal asymptote is given by \(y = -c\). This is because, as \(x\) becomes very large or very small, the term \(ab^x\) will dominate the function, and the effect of the constant \(-c\) becomes apparent.

• In the given function \(f(x) = 10^x - 2\), the \(-2\) suggests the horizontal asymptote is at \(y = -2\).
• It means as \(x\) approaches infinity, the value of \(10^x\) grows very large, causing the entire function to hover near but never actually reach \(y = -2\).

Graph Analysis

Graph analysis is the process of understanding the properties and behavior of the graph of a function. For an exponential function like \(f(x) = 10^x - 2\), you analyze different aspects to gain insight into its behavior:

• Domain and Range: The domain of \(f(x) = 10^x - 2\) is all real numbers (\(-\infty, \infty\)). The range is all real numbers above the horizontal asymptote; thus, \(y > -2\).
• Intercepts: Finding where the function crosses the axes. By setting \(x = 0\), you can find the y-intercept \(f(0) = 10^0 - 2 = 1 - 2 = -1\).
• End Behavior: Observing what happens to the graph as \(x\) approaches infinity or negative infinity. For \(10^x - 2\), as \(x\) goes to infinity, \(f(x)\) approaches but never reaches \(-2\); as \(x\) goes to negative infinity, \(f(x)\) decreases but remains above the asymptote.

Understanding these concepts helps students to interpret the graph of the function accurately, confirming that the line \(y = -2\) acts as a horizontal asymptote.