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Chapter 12 Functions and Their Graphs Their Graphs

12.1 Relations and Functions 12.2 Graphs of Functions and Transformations 12.3 Quadratic Functions and Their Graphs 12.4Applications of Quadratic Functions and Graphing Other Parabolas. 12.5The Algebra of Functions 12.6Variation 12 Functions and Their Graphs

Relations and Functions 12.1 Identify Relation, Function, Domain, and Range Let’s review some of the concepts first presented in Section 4.6, and we will extend what we have learned to include new functions.

Example 1 Identify the domain and range of each relation, and determine whether each relation is a function. Solution a). b). c). a). The domain is the set of first coordinates, The range is the set of second coordinates, “Does every first coordinate correspond to exactly one second coordinate?” YES  This relation is a function. b). The domain is The range is One of the elements in the domain, 12, corresponds to two elements in the range, 5 and 8.  This relation is not a function. c). The domain is. The range is. To determine whether this graph represents a function, recall that we can use the vertical line test. This graph fails the vertical line test.

Example 2 Determine whether each relation describes y as a function of x, and determine the domain of the relation. Solution  Now, we will look at relations and functions written as equations. a). If a value is substituted for x will have exactly one corresponding value of y. For example, if we substitute 9 for x, the only value of y is 3. Therefore, is a function. To determine the domain, ask yourself, “Is there any number that cannot be substituted for x in ?” YES, we cannot use negative numbers because square root of negative numbers is not possible. The domain consists only of positive real numbers: b). If we substitute a number such as 4 for x and solve for y, we get:  The ordered pairs (4, 2) and (4,-2) satisfy the equation. Since x=4 corresponds to two different y-values, is not a function

Graph a Linear Function Recall also that we can use function notation to represent the relationship between x and y If a linear equation in two variables y=mx +b is a function, we call it a linear function.

Example 3 a) What is the domain of f ? b) Graph the function. Solution a). The domain is the set of all real numbers that can be substituted for x. Ask yourself, “Is there any number that cannot be substituted for x in ?” No. Any real number can be substituted for x, and f(x) will be defined The domain consists of all real numbers. This can be written as b). The y-intercept is (0,2), and the slope of the line is. Use this information to graph the line.

Define a Polynomial Function and a Quadratic Function The expression is a polynomial. For each real number that is substituted for x, there will be only one value of the expression. Since each value substituted for the variable produces only one value of the expression, we can use function notation to represent a polynomial.  Another function often used in mathematics is a polynomial function. The domain of a polynomial function is all real numbers,  A Quadratic Function is a special type of polynomial function.

Evaluate Linear and Quadratic Functions for a Given Variable or Expression We can also evaluate functions for variables and expressions. Example 4a Let.Find each of the following and simplify Solution a). Finding g (m) means to substitute m for x in the function g, and simplify the expression as much as possible. Substitute m for x. g(x) = ‒ 2x+5 g(m) = ‒ 2m+5 b). Finding g ( a - 1) means to substitute a -1 for x in the function g, and simplify the expression as much as possible. Since a -1 contains two terms, we must put it in parentheses. g(a-1)= ‒ 2(a-1)+5 g(a-1)= ‒ 2a+2+5 g(a-1)= ‒ 2a+7 g(x) = ‒ 2x+5 Substitute a-1 for x. Distribute. Combine like terms.

Example 4b Let.Find each of the following and simplify Solution a). To find h(w), substitute w for x in the function h. Substitute w for x. h(x) = x 2 – x + 4 b). To find h(d+3), substitute d+3 for x in the function h, and simplify the expression as much as possible. Since d+3 contains two terms, we must put it in parentheses. Substitute d+3 for x. Distribute. Combine like terms. h(w) = w 2 – w + 4 h(x) = x 2 – x + 4 h(d+3) = (d+3) 2 – (d+3) + 4 h(d+3) = d 2 +6d + 9 ‒ d ‒ 3+ 4 h(d+3) = d 2 +5d + 10

Define and Determine the Domain of a Rational Function We can use function notation to represent a rational expression. is an example of a rational function. Since a fraction is undefined when its denominator equals zero, it follows that a rational expression is undefined when its denominator equals zero.

Example 5 Determine the domain of each rational function. Solution a). To determine the domain of f(x),set the denominator equal to zero, and solve for x. Set the denominator =0 Solve The domain contains all real numbers except -3. Write the domain in interval notation as b). To determine the domain of g(t),set the denominator equal to zero, and solve for x. Set the denominator =0 Solve Write the domain in interval notation as

Example 5-continued Determine the domain of each rational function. Solution c) Is there any number that cannot be substituted for x in h(x)? The denominator is a constant, 5, and it can never equal zero. Therefore, any real number can be substituted for x and the function will be defined. The domain consists of all real numbers, which can be written as

Define and Determine the Domain of a Square Root Function Example 6 Determine the domain of each square root function. Solution The value of the Radicand must be =0 a ). “Is there any number that cannot be substituted for x in f(x)? ” Yes. There are many values that cannot be substituted for x. Since we are considering only real numbers in the domain and range, x cannot be a negative number because then f (x) would be imaginary. Therefore, x must be greater than or equal to 0 in order to produce a real-number value for f (x). The domain consists of or b ). To determine the domain of g(r), solve the inequality: Solve Write the domain as

Summarize Strategies for Determining the Domain of a Given Function Let’s summarize what we have learned about determining the domain of a function