Title: Unit 17 Quiz
Course: Technical Math for Industry

Q: Which of the relations given by the following sets of ordered pairs is a function?
A: (a choice with no text)
If wrong: Incorrect. You may have thought the relation of this set of ordered pairs was a function because each input is only one away from each output. However, a relation is a function if there is only one output for each input. The input has two outputs: and . The relation is not a function.
If right: Correct. You could organize the ordered pairs in a table to help you see that each input in this set of ordered pairs has only one output. The relation is a function.
If wrong: Incorrect. If you organize these ordered pairs in a table, you can see that the input 0 has two outputs: 3 and . The relation is not a function.
If wrong: Incorrect. You may have thought that a relation is a function because each output has only one input. However, a relation is a function if each input has only one output. Because the input 2 in this set of ordered pairs has numerous outputs, the relation is not a function.

Q: Which function below has the following domain and range?

Domain:

Range:
A: (a choice with no text)
If right: Correct. Organize the ordered pairs in a table. The set of input values matches the domain given in the problem: . The set of output values matches the range given in the problem: . Some of the ordered pairs will have the same y-coordinate, but these values only need to be listed once in the set for the range.
If wrong: Incorrect. You may have chosen this function because the ordered pairs show the least and greatest values from the domain and the range. However, the domain is the set of all input values, and this set of input values does not match the domain given in the problem. Also, the range is the set of all output values, and this set of output values does not match the given range in the problem.
If wrong: Incorrect. You may have looked at only the first few ordered pairs in this function. However, the set of output values does not match the range given in the problem.
If wrong: Incorrect. You may have mixed up the definition of the domain and range of a function. The domain is the set of all input values, or x-coordinates, and the range is the set of all output values, or y-coordinates. The set of input values does not match the domain given in the problem, and the set of output values does not match the range given in the problem.

Q: Which graph represents a function?
A: (a choice with no text)
If wrong: Incorrect. If you draw the vertical line , you will find that it passes through two points. Therefore, the graph fails the vertical line test and does not represent a function.
If wrong: Incorrect. If you draw the vertical line , you will find that it passes through two points. Therefore, the graph fails the vertical line test and does not represent a function.
If right: Correct. Draw a vertical line through each point on the graph. Because no line intersects any other point, the graph represents a function.
If wrong: Incorrect. You may have done a horizontal line test instead of a vertical line test. If you draw a vertical line through any point on the graph, you will find that the line passes through more than one point. Therefore, the graph fails the vertical line test and does not represent a function.

Q: Given , find .
A: (a choice with no text)
If wrong: Incorrect. You substituted the expression for the entire term , instead of just for x. Your first step should be to replace x by : . Then simplify this expression.
If wrong: Incorrect. You may have correctly substituted for x to get , but you did not simplify correctly. You should first use the distributive property and then combine like terms.
If wrong: Incorrect. You multiplied the expression in the function by , instead of substituting for x. Your first step should be to replace x by : . Then simplify this expression.
If right: Correct. Substitute the expression into the function for x: Use the distributive property on the right side: Combine like terms to simplify:

Q: Which of the following is the graph of ?
A: (a choice with no text)
If wrong: Incorrect. When evaluating for negative inputs, you may have forgotten that a negative times a negative is a positive. is a linear function; its graph is a straight line.
If wrong: Incorrect. When creating a table of values for the function, you may not have taken into account the negative sign in front of the term . This graph matches the function , not .
If wrong: Incorrect. You may have chosen this graph because it crosses the x-axis at and the y-axis at , which are numbers you recognize from the function. Instead, you should create a table of values for the function, evaluating the function for different inputs and finding the related outputs.
If right: Correct. Make a table of values with inputs , 0, and 1, for example. Evaluate the function for each value of x, and record the output, . Each row in your table forms an ordered pair. These points would be , , and . The graph contains these points and is a straight line.

Q: Which of the functions is represented by the graph?
A: (a choice with no text)
If right: Correct. Make a table of values and evaluate different inputs for the function. Each row in the table is an ordered pair, which gives you a point on the graph. Some of these are , , , , and . These points are all on the graph.
If wrong: Incorrect. You may not have noticed that this answer is a linear function, so its graph would be a line. The given graph is a parabola, which is the graph of a quadratic function, or one in the form .
If wrong: Incorrect. You probably recognized that the graph is a parabola, so you chose a quadratic function with the correct vertex, . However, the graph of also contains the points and , for example, and these are not on the given graph.
If wrong: Incorrect. You probably recognized that the graph is a parabola, so you chose a quadratic function with the correct vertex, . However, the graph of also contains the points and , for example, and these are not on the given graph.

Q: Which of the following is a graph of ?
A: (a choice with no text)
If wrong: Incorrect. You confused the graph of with the graph of (shown in this answer). Subtracting 4 on the inside of the radical moves the graph to the right. Subtracting 4 on the outside moves the graph down from the graph of .
If wrong: Incorrect. This could be a graph of . When you add a number outside the radical, the graph moves up from the graph of . When you subtract a number outside the radical, the graph moves down from the graph of .
If right: Correct. Subtracting 4 from outside the radical means the graph will move four down from the graph of . The graph of will begin at and increase, as does this graph. Check further by evaluating the function for different inputs. The graph must pass through , , and , and it does.
If wrong: Incorrect. This could be a graph of . When you subtract a number outside the radical, the graph does move down from the graph of . However, that does not affect the direction in which the graph goes.

Q: Find the domain of .
- or (Correct. The value under the radical in a square root function cannot be less than 0. So the domain is all x-values for which the radicand is not negative. That means, so . This inequality is true only when x is greater than or equal to 2, or x is less than or equal to : or .)
- all real numbers (Incorrect. You may have thought that because the variable in the radical is squared, the domain includes all real numbers. However, the domain is restricted to all x-values for which the radicand is not negative. That is, . The domain is the solution of this inequality.)
- or (Incorrect. You may have incorrectly included the negative sign outside the radical when setting up the inequality to find the domain. The domain is only restricted by the value under the radical.)
A: or
If wrong: Incorrect. You may have correctly set up the inequality , but when solving the inequality, you forgot to include negative values that would also make the inequality true.

Q: What are the domain and range of the real-valued function ?
- The domain is all real numbers except 0. The range is all real numbers except 0. (Correct. The function has a variable in the denominator. The domain of such a function is restricted where the denominator is 0. In this case, that is when . Therefore, the domain is all real numbers except 0. Graph the function to find another asymptote at . This means there is no value x for which . Therefore, the range is all real numbers except 0.)
- The domain and the range are all real numbers. (Incorrect. You may have looked at the denominator and thought of a linear function, whose domain and range would be all real numbers. But this is a rational function. The denominator cannot have the value of 0. Also, if you graph the function you will find a horizontal asymptote. These two facts will help you find the domain and range.)
- The domain is all real numbers except 0. The range is all real numbers. (Incorrect. You correctly identified the domain because you knew that the denominator of the function could not be equal to 0. However, if you graph the function you will find a horizontal asymptote. This fact will help you find the range.)
- The domain is . The range is . (Incorrect. You may have known that the denominator of the function could not be equal to 0, but this does not mean that x can only have positive values. If , then for this function , so the wrong domain has led you to the wrong range. If you graph the function you will find a horizontal asymptote. This fact will help you find the range.)
A: The domain is all real numbers except 0. The range is all real numbers except 0.

Q: Find .
A: (a choice with no text)
If wrong: Incorrect. It appears that you combined the terms in the order in which they were written, combining unlike terms such as 8 with to obtain . Be sure to combine only like terms.
If wrong: Incorrect. You may have tried to simplify by first subtracting to get . However, you have used an incorrect order of operations. Note that has unlike terms and cannot be simplified. Start with: Rearrange the terms and then combine like terms.
If wrong: Incorrect. You have correctly combined like terms, but you made a mistake with one sign.
If right: Correct.

Q: Find .
A: (a choice with no text)
If wrong: Incorrect. It appears that you added the two functions instead of subtracting them. Start with: Be sure to distribute the minus sign to all terms of the second function and then combine like terms.
If wrong: Incorrect. It appears that you have made some sign errors when subtracting. Start with: Be sure to distribute the minus sign to all terms of the second function and then combine like terms.
If wrong: Incorrect. Be careful to subtract in the correct order. Start with: Distribute the minus sign to all terms of the second function and then combine like terms.
If right: Correct.

Q: Find the product of f and g.
A: (a choice with no text)
If right: Correct.
If wrong: Incorrect. You may have started with the correct first step: However, it appears that you forgot the middle terms of the product. Be sure to use FOIL when multiplying two binomials.
If wrong: Incorrect. You may have started with the correct first step: However, after using FOIL, you combined the terms incorrectly.
If wrong: Incorrect. You may have started with the correct first step: However, after using FOIL, you combined the terms incorrectly. You might have thought that combining two negative terms would give you a positive, but you were confusing addition and multiplication.

Q: Find:
A: (a choice with no text)
If wrong: Incorrect. You may have factored the denominator incorrectly. Use FOIL to check your factoring.
If wrong: Incorrect. You have found an unsimplified form of the quotient. Factor the numerator and denominator. Simplify, using common factors.
If wrong: Incorrect. You may have divided the coefficients of like terms in the numerator and denominator. Instead, start with: Factor the numerator and denominator. Simplify, using common factors.
If right: Correct.