This question is testing the concept and understanding of a function's range. It is important to recall that range can be identified graphically or algebraically. Graphically, range contains the y-values that span the image of the function where as domain, contains the x-values of the function. Algebraically, range is known as the output y of a function when input x, is used. In other words, the input values when placed into the function results in the y values creating an (x, y) pair.

For the purpose of Common Core Standards, domain and range fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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. Identify the function and what the question is asking.

Find the range of the function. 

Step 2. Discuss the options to solve the problem.

        I. Graphically plot the function by computer/technology resource. Then interpret the graph.

        II. Create a table of (x, y) pairs and plot the points to create the graph. Then interpret the graph.

        III. Algebraically find the vertex of the function and using the properties of polynomials determine the range.

For this particular function let's use the third option to find the range.

Solving the quadratic equation to find the range algebraically requires the understanding that the vertex of a parabola represents either the peak (maximum) of a function or the valley (minimum) of the function. 

Recall that a quadratic function can be written in the form,

and the formula to find the vertex x value is,

.

Finally to find the y value of the vertex we will substitute the x value found above into the original function. If the quadratic has a positive , then the vertex will occur at the valley (minimum) of the function. If  is negative then the parabola opens down thus creating a peak (maximum) of the function.

Step 3: Use algebraic technique to solve the problem.

Substituting in the values into the vertex formula is as follows.

Now, substitute the x value found into the function to find the y output value.

Step 4: Interpret the solution to answer the question.

This question is asking for the range of the function. Using the formula to find the vertex and the properties of quadratics, it was discovered that the vertex is the peak of the function. This means that all other outputs will be less than the vertex. Therefore, the range will be all real values of y for which y is less than or equal to negative five. In mathematical terms this is represented by the following.

Step 5: Verify solution by graphing the function.

Looking at the graph above, specifically the y values on the graph, range from negative five to negative infinity. Therefore, the graph verifies the solution that was found algebraically. 

This question is testing the concept and understanding of a function's domain. It is important to recall that domain can be identified graphically or algebraically. Graphically, range contains the y-values that span the image of the function where as domain, contains the x-values of the function. Algebraically, domain is known as the input values x of a function that then results in the y outputs. In other words, the input values when placed into the function results in the y values creating an (x, y) pair. It is also critical to understand that there are two important restrictions on domain. These restrictions occur when the x variable is in the denominator of a function or under a radical. This is because a fraction does not exist when there is a zero in the denominator; if under the radical is negative it results in imaginary values.

For the purpose of Common Core Standards, domain and range fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the function and what the question is asking.

Find the domain (x values) and range (y values) of the function. This includes identifying points where x values do not result in a y value. At these points the domain does not exist.

Step 2. Discuss the options to solve the problem.

        I. Graphically plot the function by computer/technology resource. Then interpret the graph.

        II. Create a table of (x, y) pairs and plot the points to create the graph. Then interpret the graph.

        III. Algebraically find the x values that do not exist, these will be areas that are not in the domain.

Any time there is a fraction set the denominator equal to zero and solve for x, this will be a value that x cannot equal since it is not in the domain. Also, anytime there is a radical set the radicand equal to zero and solve for x; this will also be a value x cannot equal as it is also not in the domain.

Fractions:

Radicals

Graphically, points where the domain of a function does not exist is seen as a vertical asymptote.

For this particular function let's use the first option to find the domain.

Interpreting the graph above, it is seen that there is a vertical asymptote at x equalling 3. Vertical asymptotes on a graph indicate that the that particular x value is not in the domain of the function. Therefore mathematically, the domain can be described as all x values greater than 3:

.

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the inside function is in the form  and the question asks to calculate  we know that the input value is two.

After the function value of  is calculated, it will be the input value for .

Step 2: Given the function, input the desired  value. 

In step one the input value of two was found. Now substitute 2 in for every in the function .

Step 3: Substitute the function value of the inside portion into .

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the function is in the form  and the question asks to calculate  we know that the input value is two.

Step 2: Given the function, input the desired  value. 

In step one the input value of two was found. Now substitute 2 in for every in the function.

Step 3: Answer the question.

This particular function when solved algebraically results in one value.

The answer is

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the inside function is in the form  and the question asks to calculate  we know that the input value is negative two.

After the function value of  is calculated, it will be the input value for .

Step 2: Given the function, input the desired  value. 

In step one the input value of negative two was found. Now substitute -2 in for every in the function .

Step 3: Substitute the function value of the inside portion into .

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the function is in the form  and the question asks to calculate  we know that the input value is negative three.

Step 2: Given the function, input the desired  value. 

In step one the input value of negative three was found. Now substitute negative three in for every  in the function.

Step 3: Answer the question.

This particular function when solved algebraically results in one value.

The answer is

.

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the function is in the form  and the question asks to calculate  we know that the input value is six.

Step 2: Given the function, input the desired  value. 

In step one the input value of six was found. Now substitute 6 in for every in the function.

Step 3: Verify solution by graphing the function.

To find the function value on a graph go to  and find the y value that lies on the graph at . Looking at the graph above when , .

Recall that , thus verifying the solution found in step 2.

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the function is in the form  and the question asks to calculate  we know that the input value is five.

Step 2: Given the function, input the desired  value. 

In step one the input value of five was found. Now substitute 5 in for every  in the function.

Step 3: Verify solution by graphing function.

To find the function value on a graph go to  and find the y value that lies on the graph at . Looking at the graph above when , .

Recall that , thus verifying the solution found in step 2.

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the function is in the form  and the question asks to calculate  we know that the input value is one.

Step 2: Given the function, input the desired  value. 

In step one the input value of one was found. Now substitute 1 in for every  in the function.

Since a fraction cannot have a zero in the denominator, the function value at  does not exist.

Step 3: Verify solution by graphing function.

Recall that graphically, a vertical asymptote represents where the function does not exist.

Looking at the above graph the function does not exist at  thus verifying the solution found in step 2.

This question is testing the knowledge and skills of calculating a function value. Similar to domain and range, calculating the function value requires the application of input values into the function to find the output value. In other words, evaluating a function at a particular value results in the function value; which is another way of saying function value is the output. 

For the purpose of Common Core Standards, function notation and evaluation fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.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: Identify the input value.

Since the function is in the form  and the question asks to calculate  we know that the input value is two.

Step 2: Given the function, input the desired  value. 

In step one the input value of two was found. Now substitute 2 in for every  in the function.

Step 3: Verify solution by graphing the function.

To find the function value on a graph go to  and find the y value that lies on the graph at . Looking at the graph above when , .

Recall that , thus verifying the solution found in step 2.