Problem 135 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.

Step-by-Step Solution

Answer

The statement does make sense, as it accurately portrays how vertical and horizontal translations affect the asymptotes of exponential and logarithmic functions, respectively. This behavior can be seen as a reflection of the inverse relationship between logarithmic and exponential functions.

Step 1: Understand Exponential and Logarithmic Functions

An exponential function is a mathematical function of the form \(f(x) = a \cdot b^x\), where 'a' and 'b' are constants, 'b' is positive and 'x' is any real number. A logarithmic function is the inverse of an exponential function. That is, if \(y = b^x\), then \(x = \log_b y\). This implies that whatever an exponential function does, a logarithmic function does the exact opposite and vice versa.

Step 2: Understand Translations or Shifts

A vertical translation or shift of a function means shifting the entire function up or down as per the factor of translation. In an exponential function, a vertical shift changes the horizontal asymptote. A horizontal translation or shift of a function means shifting the entire function left or right. In a logarithmic function, a horizontal shift changes the vertical asymptote.

Step 3: Determine the Validity of the Statement

The given statement says: 'a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote'. Given the definitions and properties discussed above, the statement indeed makes sense because it correctly describes the inverse behaviors of exponential and logarithmic functions under translations.