Select The Correct Answer.Which Logarithmic Function Has A $y$-intercept?A. $f(x)=\log (x+1)-1$ B. $f(x)=\log X+1$ C. $f(x)=\log (x-1)+1$ D. $f(x)=\log (x-1)-1$

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Introduction

Logarithmic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. In this article, we will focus on understanding logarithmic functions and their graphs, with a specific emphasis on identifying the correct answer to a question about logarithmic functions with a y-intercept.

What is a Logarithmic Function?

A logarithmic function is a mathematical function that is the inverse of an exponential function. It is defined as the power to which a base number must be raised to produce a given value. In other words, if we have an exponential function of the form y=axy = a^x, then the corresponding logarithmic function is of the form x=log⁑a(y)x = \log_a(y).

Graphs of Logarithmic Functions

The graph of a logarithmic function is a curve that is always increasing or decreasing, but never changes direction. The graph of a logarithmic function with a base greater than 1 is always increasing, while the graph of a logarithmic function with a base less than 1 is always decreasing.

Identifying the Correct Answer

Now, let's focus on the question at hand: Which logarithmic function has a y-intercept? To answer this question, we need to understand what a y-intercept is. A y-intercept is the point at which a graph intersects the y-axis. In other words, it is the value of y when x is equal to 0.

Analyzing the Options

Let's analyze each of the options given:

A. $f(x)=\log (x+1)-1$

This function has a y-intercept when x is equal to 0. Substituting x = 0 into the function, we get:

f(0)=log⁑(0+1)βˆ’1=log⁑(1)βˆ’1=0βˆ’1=βˆ’1f(0) = \log (0+1)-1 = \log (1)-1 = 0-1 = -1

So, this function has a y-intercept at the point (0, -1).

B. $f(x)=\log x+1$

This function does not have a y-intercept when x is equal to 0. Substituting x = 0 into the function, we get:

f(0)=log⁑0+1=βˆ’βˆž+1=βˆ’βˆžf(0) = \log 0+1 = -\infty +1 = -\infty

Since the function is undefined at x = 0, it does not have a y-intercept.

C. $f(x)=\log (x-1)+1$

This function does not have a y-intercept when x is equal to 0. Substituting x = 0 into the function, we get:

f(0)=log⁑(0βˆ’1)+1=log⁑(βˆ’1)+1=βˆ’βˆž+1=βˆ’βˆžf(0) = \log (0-1)+1 = \log (-1)+1 = -\infty +1 = -\infty

Since the function is undefined at x = 0, it does not have a y-intercept.

D. $f(x)=\log (x-1)-1$

This function does not have a y-intercept when x is equal to 0. Substituting x = 0 into the function, we get:

f(0)=log⁑(0βˆ’1)βˆ’1=log⁑(βˆ’1)βˆ’1=βˆ’βˆžβˆ’1=βˆ’βˆžf(0) = \log (0-1)-1 = \log (-1)-1 = -\infty -1 = -\infty

Since the function is undefined at x = 0, it does not have a y-intercept.

Conclusion

Based on our analysis, we can conclude that the only function that has a y-intercept is option A: $f(x)=\log (x+1)-1$. This function has a y-intercept at the point (0, -1).

Key Takeaways

  • A logarithmic function is a mathematical function that is the inverse of an exponential function.
  • The graph of a logarithmic function is a curve that is always increasing or decreasing, but never changes direction.
  • A y-intercept is the point at which a graph intersects the y-axis.
  • To identify the correct answer, we need to analyze each option and determine whether it has a y-intercept.

Final Answer

The final answer is option A: $f(x)=\log (x+1)-1$.