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Chapter 3: Problem 77

Determine whether the statement is true or false. Justify your answer. The line \(y=-2\) is an asymptote for the graph of \(f(x)=10^{x}-2\).

Short Answer

Expert verified
The statement 'The line \(y=-2\) is an asymptote for the graph of \(f(x)=10^{x}-2\)' is true, but it is a one-sided horizontal asymptote only for \(x\) approaching negative infinity.

Step by step solution

01

Understanding the Function

Look at the given function \(f(x)=10^{x}-2\). Here, as \(x\) approaches negative infinity, the value of \(10^{x}\) approaches 0 because any positive number raised to the power of a large negative number tends to 0. So, the value of \(f(x)\) approaches -2, which is the equation of the line in the statement.
02

Identify the Asymptote

For the function \(f(x)=10^{x} - 2\), when x approaches negative infinity, \(f(x)\) approaches -2. Hence, \(y=-2\) is an asymptote.
03

Verifying the Asymptote

For \(y=-2\) to be an asymptote, the function \(f(x)\) must approach -2 as \(x\) approaches infinity or negative infinity. As stated in Step 1, \(f(x)\) does approach -2 as \(x\) approaches negative infinity. However, as \(x\) approaches infinity, the value of \(10^{x}\) increases without bound and \(f(x)\) does not approach -2, hence \(y=-2\) is one-sided horizontal asymptote.

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