Problem 135 - College Algebra

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

Looking at the properties of exponential and logarithmic functions, both statements do make sense. In the first statement, a vertical translation would indeed shift an exponential function's horizontal asymptote. Similarly, the second statement also makes sense because a horizontal translation, indeed, shifts a logarithmic function's vertical asymptote.

Step 1: Identify Statements

The starting point would be identifying the two main statements in the exercise: 'A vertical translation shifts an exponential function's horizontal asymptote' and 'A horizontal translation shifts a logarithmic function's vertical asymptote'.

Step 2: Analyze the First Statement

A vertical translation of a function involves adding a constant to the function value which moves the graph up or down along the vertical axis. For exponential functions, \( e^x \) as an example, the horizontal asymptote is the x-axis or y=0. A vertical shift of this function would shift the horizontal asymptote up or down but would not remove its existence. Therefore, this statement makes sense.

Step 3: Analyze the Second Statement

A horizontal translation of a function involves adding a constant to the input value (x) which moves the graph left or right along the horizontal axis. For logarithmic functions, such as \( \log x \), the vertical asymptote is the y-axis or x=0. A horizontal shift of this function would shift the vertical asymptote left or right still maintaining its presence. Therefore, this statement is sensical.