§ 2.2 Graphs of Functions

The graph of a function is just the graph of its ordered pairs. Graphs of Functions For example, the graph of y = 3x is the set of points (x, y) satisfying y = 3x. The graph of a function is just the graph of its ordered pairs. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.2

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

Graphs of Functions CONTINUED Blitzer, Intermediate Algebra, 5e – Slide #4 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, Intermediate Algebra, 5e – Slide #5 Section 2.2

The Vertical Line Test EXAMPLE Use the vertical line test to identify graphs in which y is a function of x. (a) (b) (c) Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.2

The Vertical Line Test (a) (b) (c) CONTINUED SOLUTION y is a function of x y is not a function of x y is a function of x Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.2

The Vertical Line Test (b) CONTINUED SOLUTION 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. CONTINUED SOLUTION (b) y is not a function of x Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.2

Graphs of Functions EXAMPLE 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, Intermediate Algebra, 5e – Slide #9 Section 2.2

Graphs of Functions CONTINUED (a) Find f (3). What does this mean in terms of the variables in this situation? (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, Intermediate Algebra, 5e – Slide #10 Section 2.2

Graphs of Functions CONTINUED SOLUTION (a) Find f (3). What does this mean in terms of the variables in this situation? f (3) = 0.83. This means that when a first-class letter weighs 3 ounces, postage costs 83 cents. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.2

Graphs of Functions CONTINUED (b) Find f (4). What does this mean in terms of the variables in this situation? f (4) = 1.06. This means that when a first-class letter weighs 4 ounces, postage costs $1.06. Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.2

Graphs of Functions CONTINUED (c) What is the cost of mailing a letter that weighs 1.5 ounces? f (1.5) = 0.60. This means that when a first-class letter weighs 1.5 ounces, postage costs $0.60. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.2

Graphs of Functions CONTINUED (d) What is the cost of mailing a letter that weighs 1.8 ounces? f (1.8) = 0.60. This means that when a first-class letter weighs 1.8 ounces, postage costs $0.60. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.2

Obtaining Information from Graphs 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, Intermediate Algebra, 5e – Slide #15 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, Intermediate Algebra, 5e – Slide #16 Section 2.2

Domain and Range EXAMPLE Use the graph of the function to identify its domain and range. Blitzer, Intermediate Algebra, 5e – Slide #17 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, Intermediate Algebra, 5e – Slide #18 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, Intermediate Algebra, 5e – Slide #19 Section 2.2

Domain and Range Domain = {x | x is a real number} Range = {y | y 3.6} SOLUTION Domain = {x | x is a real number} Range = {y | y 3.6} Blitzer, Intermediate Algebra, 5e – Slide #20 Section 2.2