This question is testing ones ability to understand what it means for a function to be invertible or non-invertible and how to find the inverse of a non-invertible function through means of domain restriction.

For the purpose of Common Core Standards, "Produce an invertible function from a non-invertible function by restricting the domain." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.4d). It is important to note that this standard is not directly tested on but, it is used for building a deeper understanding on invertible and non-invertible functions and their inverses.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Determine whether the function given is invertible or non-invertible.

Using technology to graph the function  results in the following graph.

This function is non-invertible because when taking the inverse, the graph will become a parabola opening to the right which is not a function. A sideways opening parabola contains two outputs for every input which by definition, is not a function.

Step 2: Make the function invertible by restricting the domain.

To make the given function an invertible function, restrict the domain to  which results in the following graph.

Step 3: Graph the inverse of the invertible function.

Swapping the coordinate pairs of the given graph results in the inverse.

Therefore, the inverse of this function algebraically is

This question is testing ones ability to understand what it means for a function to be invertible or non-invertible and how to find the inverse of a non-invertible function through means of domain restriction.

For the purpose of Common Core Standards, "Produce an invertible function from a non-invertible function by restricting the domain." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.4d). It is important to note that this standard is not directly tested on but, it is used for building a deeper understanding on invertible and non-invertible functions and their inverses.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Determine whether the function given is invertible or non-invertible.

Using technology to graph the function  results in the following graph.

This function is non-invertible because when taking the inverse, the graph will become a parabola opening to the right which is not a function. A sideways opening parabola contains two outputs for every input which by definition, is not a function.

Step 2: Make the function invertible by restricting the domain.

To make the given function an invertible function, restrict the domain to  which results in the following graph.

Step 3: Graph the inverse of the invertible function.

Swapping the coordinate pairs of the given graph results in the inverse.

Therefore, the inverse of this function algebraically is

This question is testing ones ability to understand what it means for a function to be invertible or non-invertible and how to find the inverse of a non-invertible function through means of domain restriction.

For the purpose of Common Core Standards, "Produce an invertible function from a non-invertible function by restricting the domain." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.4d). It is important to note that this standard is not directly tested on but, it is used for building a deeper understanding on invertible and non-invertible functions and their inverses.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Determine whether the function given is invertible or non-invertible.

Using technology to graph the function  results in the following graph.

This function is non-invertible because when taking the inverse, the graph will become a parabola opening to the right which is not a function. A sideways opening parabola contains two outputs for every input which by definition, is not a function.

Step 2: Make the function invertible by restricting the domain.

To make the given function an invertible function, restrict the domain to  which results in the following graph.

Step 3: Graph the inverse of the invertible function.

Swapping the coordinate pairs of the given graph results in the inverse.

Therefore, the inverse of this function algebraically is

This question is testing ones ability to understand what it means for a function to be invertible or non-invertible and how to find the inverse of a non-invertible function through means of domain restriction.

For the purpose of Common Core Standards, "Produce an invertible function from a non-invertible function by restricting the domain." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.4d). It is important to note that this standard is not directly tested on but, it is used for building a deeper understanding on invertible and non-invertible functions and their inverses.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Determine whether the function given is invertible or non-invertible.

Using technology to graph the function  results in the following graph.

This function is non-invertible because when taking the inverse, the graph will become a parabola opening to the right which is not a function. A sideways opening parabola contains two outputs for every input which by definition, is not a function.

Step 2: Make the function invertible by restricting the domain.

To make the given function an invertible function, restrict the domain to  which results in the following graph.

Step 3: Graph the inverse of the invertible function.

Swapping the coordinate pairs of the given graph results in the inverse.

Therefore, the inverse of this function algebraically is

This question is testing ones ability to understand what it means for a function to be invertible or non-invertible and how to find the inverse of a non-invertible function through means of domain restriction.

For the purpose of Common Core Standards, "Produce an invertible function from a non-invertible function by restricting the domain." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.4d). It is important to note that this standard is not directly tested on but, it is used for building a deeper understanding on invertible and non-invertible functions and their inverses.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Determine whether the function given is invertible or non-invertible.

Using technology to graph the function  results in the following graph.

This function is non-invertible because when taking the inverse, the graph will become a parabola opening to the right which is not a function. A sideways opening parabola contains two outputs for every input which by definition, is not a function.

Step 2: Make the function invertible by restricting the domain.

To make the given function an invertible function, restrict the domain to  which results in the following graph.

Step 3: Graph the inverse of the invertible function.

Swapping the coordinate pairs of the given graph results in the inverse.

Therefore, the inverse of this function algebraically is

This question is testing ones ability to understand what it means for a function to be invertible or non-invertible and how to find the inverse of a non-invertible function through means of domain restriction.

For the purpose of Common Core Standards, "Produce an invertible function from a non-invertible function by restricting the domain." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.4d). It is important to note that this standard is not directly tested on but, it is used for building a deeper understanding on invertible and non-invertible functions and their inverses.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Determine whether the function given is invertible or non-invertible.

Using technology to graph the function  results in the following graph.

This function is non-invertible because when taking the inverse, the graph will become a parabola opening to the right which is not a function. A sideways opening parabola contains two outputs for every input which by definition, is not a function.

Step 2: Make the function invertible by restricting the domain.

To make the given function an invertible function, restrict the domain to  which results in the following graph.

Step 3: Graph the inverse of the invertible function.

Swapping the coordinate pairs of the given graph results in the inverse.

Therefore, the inverse of this function algebraically is

This question is testing one's ability to identify and prove whether situations and their functions are linear or exponential. The key concept in questions like these, is understanding and recognizing that linear functions grow by equal differences over intervals, where as exponential functions grow by equal factors over intervals.

For the purpose of Common Core Standards, proving that linear functions grow by difference and exponential functions grow by factors, falls within the Cluster A of construct and compare linear, quadratic, and exponential model and solve problems concept (CCSS.Math.content.HSF.LE.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Examine the first statement.

   I. Identify if it is increase/decreasing by a constant or by a percentage that is dependent on another value in the statement.

      Statement one: Tina's allowance increases  every month. In this particular case the amount of increase depends on the monthly allowance. In other words as the monthly allowance increases the  increase becomes a greater dollar amount as time increases. Therefore, the increase is by a percentage. 

   II. Conclude whether the statement is linear or exponential.

      Since the increase is by a percentage, statement one is an exponential function.

Step 2: Examine the second statement.

   I. Identify if it is increase/decreasing by a constant or by a percentage that is dependent on another value in the statement.

      Statement two: Johnny is  years older than Jane. In this particular case Johnny's age can be written as a function of Jane's age. Since Johnny is  years older than Jane this is a constant difference. In other words, as the years go on, Johnny's age increase but so does Jane's therefore, Johnny remains  years older than Jane across all time intervals.

   II. Conclude whether the statement is linear or exponential.

      Since the increase is by a constant, statement two is a linear function.

Step 3: Answer the question.

Tina's allowance increases  every month is an exponential function.

Johnny is  years older than Jane is a linear function.

Recall that for a statement to be linear it must show growth of a constant difference. This means that the output has the same difference for each one increase to the input value. For a statement to be exponential, it must show growth by a common factor.

This question is testing one's ability to identify and prove whether situations and their functions are linear or exponential. The key concept in questions like these, is understanding and recognizing that linear functions grow by equal differences over intervals, where as exponential functions grow by equal factors over intervals.

For the purpose of Common Core Standards, proving that linear functions grow by difference and exponential functions grow by factors, falls within the Cluster A of construct and compare linear, quadratic, and exponential model and solve problems concept (CCSS.Math.content.HSF.LE.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Examine the statement.

   I. Identify if it is increase/decreasing by a constant or by a percentage that is dependent on another value in the statement.

      Statement: Tina's allowance increases by  every month. In this particular case the amount of increases by a constant of two dollars every month. 

   II. Conclude whether the statement is linear or exponential.

      Since the increase is by a constant, the statement is a linear function.

Step 3: Answer the question.

Tina's allowance increases by  every month is a linear function.

Recall that for a statement to be linear it must show growth of a constant difference. This means that the output has the same difference for each one increase to the input value. For a statement to be exponential, it must show growth by a common factor.

This question is testing one's ability to identify and prove whether situations and their functions are linear or exponential. The key concept in questions like these, is understanding and recognizing that linear functions grow by equal differences over intervals, where as exponential functions grow by equal factors over intervals.

For the purpose of Common Core Standards, proving that linear functions grow by difference and exponential functions grow by factors, falls within the Cluster A of construct and compare linear, quadratic, and exponential model and solve problems concept (CCSS.Math.content.HSF.LE.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Examine the statement.

   I. Identify if it is increase/decreasing by a constant or by a percentage that is dependent on another value in the statement.

      Statement: Tina's allowance increases  every month. In this particular case the amount of increase depends on the monthly allowance. In other words as the monthly allowance increases the  increase becomes a greater dollar amount as time increases. Therefore, the increase is by a percentage. 

   II. Conclude whether the statement is linear or exponential.

      Since the increase is by a percentage, statement one is an exponential function.

Step 3: Answer the question.

Tina's allowance increases  every month is an exponential function.

Recall that for a statement to be linear it must show growth of a constant difference. This means that the output has the same difference for each one increase to the input value. For a statement to be exponential, it must show growth by a common factor.

This question is testing one's ability to identify and prove whether situations and their functions are linear or exponential. The key concept in questions like these, is understanding and recognizing that linear functions grow by equal differences over intervals, where as exponential functions grow by equal factors over intervals.

For the purpose of Common Core Standards, proving that linear functions grow by difference and exponential functions grow by factors, falls within the Cluster A of construct and compare linear, quadratic, and exponential model and solve problems concept (CCSS.Math.content.HSF.LE.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Examine the statement.

   I. Identify if it is increase/decreasing by a constant or by a percentage that is dependent on another value in the statement.

      Statement: Johnny is  years older than Jane. In this particular case Johnny's age can be written as a function of Jane's age. Since Johnny is  years older than Jane this is a constant difference. In other words, as the years go on, Johnny's age increase but so does Jane's therefore, Johnny remains  years older than Jane across all time intervals.

   II. Conclude whether the statement is linear or exponential.

      Since the increase is by a constant, statement two is a linear function.

Step 3: Answer the question.

Johnny is  years older than Jane is a linear function.

Recall that for a statement to be linear it must show growth of a constant difference. This means that the output has the same difference for each one increase to the input value. For a statement to be exponential, it must show growth by a common factor.