§ 2.2 Graphs of Functions

Blitzer, Algebra for College Students, 6e – Slide #2 Section 2.2 Graphs of Functions The graph of a function is just the graph of its ordered pairs. For example, the graph of y = 3x is the set of points (x, y) satisfying y = 3x.

Blitzer, Algebra for College Students, 6e – Slide #3 Section 2.2 Graphs of FunctionsEXAMPLE SOLUTION Graph the function. xy = -3x+1Ordered Pair (x,y) -2f(x) = -3(-2) + 1 = = 7(-2,7) f(x) = -3(-1) + 1 = = 4(-1,4) 0f(x) = -3(0) + 1 = = 1(0,1) 1f(x) = -3(1) + 1 = = -2(1,-2) 2f(x) = -3(2) + 1 = = -5(2,-5)

Blitzer, Algebra for College Students, 6e – Slide #4 Section 2.2 Graphs of FunctionsCONTINUED

Blitzer, Algebra for College Students, 6e – Slide #5 Section 2.2 The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. (a)(b)(c)

Blitzer, Algebra for College Students, 6e – Slide #6 Section 2.2 The Vertical Line TestEXAMPLE Use the vertical line test to identify graphs in which y is a function of x. (a)(b)(c)

Blitzer, Algebra for College Students, 6e – Slide #7 Section 2.2 The Vertical Line TestSOLUTION CONTINUED (a)(b)(c) y is a function of xy is not a function of xy is a function of x

Blitzer, Algebra for College Students, 6e – Slide #8 Section 2.2 The Vertical Line TestSOLUTION CONTINUED (b) y is not a function of x This graph is not a function since the blue vertical line indicated picked up three points of the graph. Here, three values of y correspond to one value of x. For example, the points indicated might have been (-2,4), (-2,0) and (-2,-4) and the x value of -2 then mapped to three distinct y values and not just one. Whatever the exact values are, in this graph it is clear that y is not a function of x. The vertical line test is just a quick visual method for determining whether you have a function.

Blitzer, Algebra for College Students, 6e – Slide #9 Section 2.2 Graphs of FunctionsEXAMPLE The figure shows the cost of mailing a first-class letter, f(x), as a function of its weight, x, in ounces. Use the graph to answer the following questions.

Blitzer, Algebra for College Students, 6e – Slide #10 Section 2.2 Graphs of Functions (a) Find f (3). What does this mean in terms of the variables in this situation? CONTINUED (b) Find f (4). What does this mean in terms of the variables in this situation? (c) What is the cost of mailing a letter that weighs 1.5 ounces? (d) What is the cost of mailing a letter that weighs 1.8 ounces?

Blitzer, Algebra for College Students, 6e – Slide #11 Section 2.2 Graphs of FunctionsSOLUTION CONTINUED f (3) = This means that when a first-class letter weighs 3 ounces, postage costs 83 cents. (a) Find f (3). What does this mean in terms of the variables in this situation?

Blitzer, Algebra for College Students, 6e – Slide #12 Section 2.2 Graphs of FunctionsCONTINUED f (4) = This means that when a first-class letter weighs 4 ounces, postage costs $1.06. (b) Find f (4). What does this mean in terms of the variables in this situation?

Blitzer, Algebra for College Students, 6e – Slide #13 Section 2.2 Graphs of FunctionsCONTINUED f (1.5) = This means that when a first-class letter weighs 1.5 ounces, postage costs $0.60. (c) What is the cost of mailing a letter that weighs 1.5 ounces?

Blitzer, Algebra for College Students, 6e – Slide #14 Section 2.2 Graphs of FunctionsCONTINUED f (1.8) = This means that when a first-class letter weighs 1.8 ounces, postage costs $0.60. (d) What is the cost of mailing a letter that weighs 1.8 ounces?

Blitzer, Algebra for College Students, 6e – Slide #15 Section 2.2 Graphs of Functions Obtaining Information from Graphs A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph. An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph. An arrow indicates that the graph extends indefinitely in the direction in which the arrow points.

Blitzer, Algebra for College Students, 6e – Slide #16 Section 2.2 Domain and Range The graph of a function can be used to determine the function’s domain and range. Domain: set of inputs (found on the x axis – the collection of all x values in the graph) Range: set of outputs (found on the y axis – the collection of all y values in the graph)

Blitzer, Algebra for College Students, 6e – Slide #17 Section 2.2 Domain and RangeEXAMPLE Use the graph of the function to identify its domain and range.

Blitzer, Algebra for College Students, 6e – Slide #18 Section 2.2 Domain and Range EXAMPLE, Continued To identify the domain, we look from the far left to the far right, identifying all the x values used. There is not a first (smallest) and there is not a last (largest) x value (indicated by the arrows). Therefore, x takes on all values.

Blitzer, Algebra for College Students, 6e – Slide #19 Section 2.2 Domain and Range EXAMPLE, Continued To identify the range, we look from the bottom of the graph to the top, identifying all the y values used. There is not a lowest, (as indicated again by the arrows) but there is a highest. Y takes on all values up to and including approximately 3.6.

Blitzer, Algebra for College Students, 6e – Slide #20 Section 2.2 Domain and Range {x | x is a real number} SOLUTION {y | y 3.6} Domain = Range =

2.2 Assignment p.111 (2-10 even, all, even, 41)