Which properties are present in a table that represents an exponential function in the form [tex]$y=b^x$[/tex] when [tex]$b=1$[/tex]?

I. As the [tex]$x$[/tex]-values increase, the [tex]$y$[/tex]-values increase.
II. The point [tex]$(1,0)$[/tex] exists in the table.
III. As the [tex]$x$[/tex]-values increase, the [tex]$y$[/tex]-values decrease.
IV. As the [tex]$x$[/tex]-values decrease, the [tex]$y$[/tex]-values decrease, approaching a singular value.

A. I and IV
B. I and II
C. II and III
D. III only

The function y = 1 x simplifies to y = 1 for all x .
Statement I is false because y remains constant as x increases.
Statement II is false because the point ( 1 , 1 ) exists, not ( 1 , 0 ) .
Statement III is false because y remains constant as x increases.
Statement IV is false because y remains constant as x decreases, not approaching a singular value.
Therefore, none of the provided options are correct.

Explanation

Understanding the Function Let's analyze the properties of the exponential function y = b x when b = 1 . In this case, the function becomes y = 1 x . This means that for any value of x , y will always be equal to 1, since 1 raised to any power is 1.

Evaluating the Statements Now, let's evaluate each statement:


I. As the x -values increase, the y -values increase. This statement is false because as x increases, y remains constant at 1.
II. The point ( 1 , 0 ) exists in the table. This statement is false because when x = 1 , y = 1 1 = 1 . So, the point ( 1 , 1 ) exists, not ( 1 , 0 ) .
III. As the x -values increase, the y -values decrease. This statement is false because as x increases, y remains constant at 1.
IV. As the x -values decrease, the y -values decrease, approaching a singular value. This statement is also false because as x decreases, y remains constant at 1. It does not approach any other value.

Checking the Options Based on the analysis, none of the individual statements are true. However, we need to find the combination of statements that is correct. Let's re-examine the options:


I and IV: Both statements are false.
I and II: Both statements are false.
II and III: Both statements are false.
III only: Statement III is false.


Conclusion Since none of the statements are individually true, and none of the combinations of statements are true, it seems there might be an issue with the question or the provided options. However, based on our analysis, we can conclude that none of the provided options accurately describe the properties of the function y = 1 x .

Examples
Consider a scenario where you have a machine that always produces 1 unit of output, regardless of the input. This can be modeled by the function y = 1 x , where x is the input and y is the output. No matter how much input you provide ( x ), the output ( y ) remains constant at 1. This concept is useful in understanding constant functions and their applications in various fields.

Answered by GinnyAnswer | 2025-07-08

All provided statements regarding the properties of the function y = 1 x are false because the function always returns 1 for all values of x . Therefore, none of the answer options are correct.
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Answered by Anonymous | 2025-08-13