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• 1. MATH: PRE-CALCULUS Section IV DePaul Math Placement Test
• 2.
• Functions
• In order to discuss functions, you need to understand of their basic characteristics.
• A function describes a relationship between one or more inputs and one output. The inputs to a function are variables. The output of the function for a particular value of x is usually represented as f(x) or g(x). When a function of a single variable is graphed on the (x,y) plane, the output of the function, f(x), is graphed on the y-axis; functions are therefore commonly written as y = x 2 rather then f(x) = x 2 .
• Two main characteristics of functions:
• The domain is the set of inputs (x values) for which the function is defined.
• Consider the following two functions: f(x)= x 2 and g(x)= 1 ⁄ x . In f(x), any value of x
• can produce a valid result since any number can be squared. In g(x), though, not
• every value of x can generate an output: when x = 0, g(x) is undefined.
• The range of a function is closely related to the domain. Whereas the domain
• is the set of inputs that a function can take, the range is the set of outputs
• that a function can produce.
• Try to think of all the values that can be generated when a number is squared. Well,
• all squares are positive (or equal to 0 ), so f(x) can never be negative. In the case of g(x), almost every number is part of the range. In fact, the only number that cannot
• be generated by the function g(x) is 0 . Therefore the range of the function g(x) is all numbers except zero.
•
• 3.
• Domain of a Function
•   We discussed the concept of a function in the previous section . The domain of a function is the set of inputs to the function that produce valid outputs. It is common for a domain to include only positive numbers, only negative numbers, or even all numbers except one or two points. As an example of a function that is undefined on a certain interval, consider f(x) = . A negative number has no square root defined in the real number system, so f(x) is undefined for all x < 0 .
•
• Finding the Domain of a Function
• When you are solving a problem you should begin by assuming that the domain is the set of real numbers.
• Next you should look for any restrictions on the domain. For example, in the case of f(x) = . We must restrict the domain to non-negative numbers since we know that you can’t take the square root of a negative number.
• There are two main restrictions to be on the lookout for:
• 1. Division by zero. Division by zero is mathematically impossible. A function is therefore undefined for all the values of x for which division by 0 occurs. For example, f(x) = 1 ⁄ x – 2 is undefined at x = 2, since when x = 2, the function is equal to f(x) = 1 ⁄ 0 .
• 2. Even roots. An even root (a square root, fourth root, etc.) of a negative number does not exist. A function is undefined for all values of x that cause a negative number to be the radicand of an even root.
•
• 4.
• Evaluating Functions
Evaluating a function simply means finding f(x) at some specific value x. Here you will have to evaluate a function at some particular constant. Take a look at the following example: If f(x) = x 2 – 3 , what is f(5)? Evaluating a function at a constant involves nothing more than substituting the constant into the definition of the function. In this case, substitute 5 for x: f(5) = 5 2 -3 = 22   In the Placement test there might be questions based on how to evaluate a function at a variable rather than a constant. Q. If f(x) = 3x / 4–x , what is f(x + 1)? A. To solve problems of this sort, follow the same method you used for evaluating a function at a constant: substitute the variable into the equation. To solve the sample question, substitute x + 1) for x in the definition of the function: f(x + 1)=   f(x+1) =   f(x+1) =
• 5.
• Operations on Functions
Functions can be added, subtracted, multiplied, and divided like any other quantity. There are a few rules that make these operations easier. For any two functions f(x) and g(x): Rule Example Addition (f+g)(x)=f(x) +g(x) If f(x) = sin x, and g(x) = cos x: (f + g)(x) = sin x + cos x Subtraction (f-g)(x)=f(x) -g(x) If f(x) = x 2 + 5, and g(x) = x 2 + 2x + 1: (f – g)(x) = x 2 + 5 – x 2 – 2x – 1 = –2x + 4 Multiplication (f X g)(x)=f(x) X g(x) If f(x) = x, and g(x) = x 3 + 8: (f+g)(x)=x(x 3 +8)= x 4 +8x Division (f ÷ g)(x)=f(x) If g(x) 0 If f(x) = 2 x and g(x) = 2 x 2 (f g)(x)=f(x) g(x) = 2x/x 2 = 1/x. Here you have to be aware of possible situations in which you inadvertently divide by zero. Since division by zero is not allowed , you should remember that any time you are dividing functions, like f (x) ⁄ g(x) , the result of the function is undefined.
• 6.
• Recognizing Functions
Recognizing that these two situations cause the function to be undefined is the key to finding any restriction on the function’s domain . Once you’ve discovered where the likely problem spots are, you can usually find the values to be eliminated from the domain easily. Q. What is the domain of f(x)= ? A. In this question , f(x) has variables in its denominator, which should be a red flag that alerts you to the possibility of division by zero. We may need to restrict the function’s domain to ensure that division by zero does not occur. To find the values of x that cause the denominator to equal zero, set up an equation and factor the quadratic: x 2 + 5x + 6 = (x + 2)(x + 3) = 0. For x = {–2, –3}, the denominator is zero and f(x) is undefined. Since it is defined for all other real numbers, the domain of f(x) is the set of all real numbers x such that x ≠ –2, –3 . This can also be written as {x: x ≠ –2, –3}.
• 7.
• Recognizing Functions
• Here’s another example:
• Q. What is the domain of f ( x ) = ?
• A. This function has both warning signs: an even root and a variable in the denominator. It’s best to examine each situation separately:
• 1.The denominator would equal zero if x = 7.
• 2.The quantity under the square root (the radicand ), x – 4 , must be greater than or equal to zero in order for the function to be defined. Therefore, x ≥ 4 .
• The domain of the function is therefore the set of real numbers x such that x ≥ 4, x ≠ 7.
• 8.
• Inverse Functions
• The inverse of a function “undoes” that function. An example may be the best way to explain what this means: the inverse of x 2 is. Let’s see how “undoes” x 2 :
•   x =10
• 10 2 =100
• = 10
• For the placement test it is important to know how to find the inverse of a simple function mathematically.
• The following rules will always work to find the inverse of a function
• Replace the variable f(x) with y.
• Switch the places of x and y .
• Solve for y.
• Replace y with f –1 (x).
• 9.
• Inverse Functions
What is the inverse of the function f(x) = ? First, replace f(x) with y. Then switch the places of x and y, and solve for y.   x = x 2 = 5x 2 = 2y 2 – 3   5x 2 + 3= 2y 2   (5/2)x 2 + (3/2) =y 2   y=   f –1 (x)=
• 10.
• Verification-Inverse of a Function is a Function
• Contrary to their name, inverse functions are not necessarily functions at all . Take a look at this question:
• Is the inverse of f(x) = x 3 a function?
• Begin by writing y = x 3 .
• switch the places of x and y: x = y 3 . Solve for y: y = 3.
• Now you need to analyze the inverse of the function and decide whether for every x, there is only one y.
• If only one y is associated with each x, you’ve got a function. Otherwise, you don’t. In this case, every x value that falls within the domain x≥0, turns out one value for y, so f –1 (x) is a function.
•
• 11.
• Verification:Inverse of a Function is a Function
Here’s another question: Q .What is the inverse of f(x) = 2|x – 1|, and is it a function? A. Replace x with y and solve for y: x = 2|y – 1| 1 x =|y-1| 2   Now, since you’re dealing with an absolute value, split the equations : y-1 =1/2 x or 1-y=1/2x Therefore, y=1/2x +1 or y = -1/2x + 1   The inverse of f(x) is this set of two equations. As you can see, for every value of x except 0, the inverse of the function assigns two values of y. Consequently, f –1 (x) is not a function.
• 12.
• Compound Functions
A compound function is a function that operates on another function . A compound function is written as nested functions, in the form f(g(x)). To evaluate a compound function, first evaluate the internal function , g(x). Next, evaluate the outer function at the result of g(x). Work with the inner parentheses first and then the outer ones, just as in any other algebraic expression. Here’s an example: Suppose h(x) = x 2 + 2x and j(x) = | (x /4) + 2|. What is j(h(4))? To evaluate this compound function, first evaluate h(4): h(4) =4 2 +2(4) =24   Now plug 24 into the definition of j: j(24)=| 24/4+2| j(24)=8 It is important that you pay attention to the order in which you evaluate the compound function. Always evaluate the inner function first . For example, if we had evaluated j(x) b efore h(x) in the above question, you would get a completely different answer: h(j(4))=h(|4/4 +2|) =h(|1+2|) =h(3) =15
• 13.
• Compund Functions
Here Suppose f(x) = 3x + 1 and g(x) = . What is g(f(x))? When you are not given a constant at which to evaluate a compound function, you should simply substitute the definition of f ( x ) as the input to g ( x ). This situation is exactly the same as a regular equation being evaluated at a variable rather than a constant. g(f(x)) =g(3x+1) = =
• 14.
• Range and Domain in Graphing
The range and domain of a function are easy enough to see in their graphs. The domain is the set of all x -values for which the function is defined. The range is the set of all y -values for which the function is defined. To find the domain and range of a graph, just look at which x - and y -values the graph includes.   Certain kinds of graphs have specific ranges and domains that are visible in their graphs. A line whose slope is not 0 (a horizontal line) or undefined (a vertical line) has the set of real numbers as its domain and range. Since a line, by definition, extends infinitely in both directions, it passes through all possible values of x and y :
• 15.
• Domain of a Function
• A n odd-degree polynomial , which is a polynomial whose highest degree of power is an odd number, also has the set of real numbers as its domain and range:
• An even-degree polynomial, which is a polynomial whose highest degree of power is an even number, has the set of real numbers as its domain, but it has a restricted range. The range is usually bounded at one end and unbounded at the other. The following parabola has range {–∞, 2 }:
• An Odd degree Polynomial Graph An even degree polynomial graph
•
• 16.
• Domain of a Function
The vertical line test makes sense because the definition of a function requires that any x -value have only one y -value. A vertical line has the same x -value along the entire line; if it intersects the graph more than once, then the graph has more than one y -value associated with that x -value. Using the vertical line test, check to see that the three graphs below are functions. The next three graphs are not functions. In each graph, a strategically placed vertical line (depicted by the dashed line) will intersect the graph more than once.
• 17.
• Graphing Functions
While most of the function questions on the Math placement test involves analysis and manipulation of the functions themselves, you will sometimes be asked a question about the graph of a function. A common question of this type asks you to match a function’s graph to its definition. Understanding the next few topics will help prepare you for questions relating to functions and their graphs.   Identifying Whether a Graph is a Function For Placement test, it is important to be able to determine if a given graph is indeed a function. A foolproof way to do this is to use the vertical line test: if a vertical line intersects a graph more than once, then the graph is not a function.   The vertical line test makes sense because the definition of a function requires that any x - value have only one y - value. A vertical line has the same x -value along the entire line; if it intersects the graph more than once, then the graph has more than one y - value associated with that x - value. Using the vertical line test, check to see that the three graphs below are functions. Continued next page.
• 18.
• Graphing Function
Trigonometric functions have various domains and ranges depending on the function. Sine, for example, has the real numbers for its domain and {–1, 1} for its range. A more detailed breakdown of the domains and ranges for the various trigonometric functions can be found in the Trigonometry chapter. Some functions have limited domains and ranges that cannot be categorized simply, but are still obvious to see. By looking at the graph, you can see that the function below has domain {3, ∞} and range {–∞, –1}. A Trigonometric function graph A limited domain and range graph
• 19.
• Logarithms
Logarithms are closely related to exponents and roots. A logarithm is the power to which you must raise a given number, called the base, to equal another number. For example, log 2 8 = 3 because 2 3 = 8 . In this case , 2 is the base and 3 is the exponent. Having defined logarithms in a sentence, let’s show one symbolically. The next two equations are equivalent: log a x = b as a b = x log 10 1000 = 3 because 10 3 =1000 log 4 ¼ = 2 because (1/2) 2 = ¼ For example, log 4 16 = 2 because 4 2 = 16 and = 4. You should now be able to see why the three topics of exponents, roots, and logarithms are often linked together. Each method provides a way to isolate one of the three variables in these types of equations. In the example above, a is the base, b is the exponent, and x is the product. Finding the root, logarithm, and exponent isolates these values, respectively.
• 20.
• Operations on Logarithms
• You will rarely see a test question involving basic logarithms such as log 10 100, or log 2 4. In particular, on the logarithm questions you’ll see in the Algebra chapter, you’ll need to be able to manipulate logarithms within equations. So, you should know how to perform the basic operations on logarithms:
• The Product Rule: when logarithms of the same base are multiplied, the base remains the same, and the exponents can be added.
• log x jk= log x j +log x k { Example: log 2 4= log 2 3 +log 2 12}
•
• The Quotient Rule : when logarithms of the same base are divided, the exponents must be subtracted.
• log x (j/k)= log x j – log x k {Example: log x (1/2)= log x 1 - log x 2}
• The Power Rule: when a logarithm is raised to a power, the exponent can be brought in front and multiplied by the logarithm.
• log x c n = n X log x c {Example: log x= 4 X logx}
•
• You might have noticed how similar these rules are to those for exponents and roots. This similarity results from the fact that logarithms are just another way to express an exponent.
• 21.
• Exponential (e) and natural Log (ln)
Exponential Functions - The exponential function is a function in mathematics. The application of this function to a value x is written as exp(x) . Equivalently, this can be written in the form e x , where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828. Natural Logarithms- The natural logarithm , the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828. In simple terms, the natural logarithm ( ln) of a number x is the power to which e would have to be raised to equal x — for example the natural log of e itself is 1 because e 1 = e, while the natural logarithm of 1 would be 0 , since e 0 = 1   NOTE- All the laws of the of logarithms with base 10 can be applied to all natural logarithm The relationship between exponential function a x (called the exponential function with base a ) is defined using the natural logarithms as follows:   e ln(x) = x if x>0 ln(e x ) = x.   And, a x =( e lna ) x = e x ln a Now, test your knowledge of the topics discussed by clicking on the sample test links given below.