College Algebra: Lesson 1 Functions and Graphs

Topics Topics include basics of functions and their graphs linear functions and slope transformations of functions combinations of functions/composite functions inverse functions

Relations A relation is any set of ordered pairs. {(1,2), (3,4), (5,6)} The domain comprises the set of first elements. {1, 3, 5} The range comprises the set of second elements. {2, 4, 6}

Functions A function is a relation for which each element in the domain corresponds to exactly one element in the range. A function can have two different first components with the same second component. {(1, 2), (3, 2), (4, 1)} A function cannot have the same first component and different second components. {(1, 2), (3, 2), (1, 3)}

Functions Functions are usually expressed in terms of equations involving x and y where x is the independent variable and y is the dependent variable. If an equation defines a function, the value of the function at x, f(x), often replaces y.

Example y = 2x + 6 Evaluate for x = 3 f(x) = 2x + 6 Evaluate for x = 3 Equation Functional Notation y = 2x + 6 Evaluate for x = 3 y = 2(3) + 6 = 12 f(x) = 2x + 6 Evaluate for x = 3 f(3) = 2(3) + 6 = 12 Example

Graphs of functions The graph of a function is the graph of its ordered pairs. Not every graph is the graph of a function.

Vertical Line Test If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

Relation Function Example

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.

Obtaining Information from Graphs To find the domain, look for all of the inputs on the x-axis that correspond to points on the graph. To find the range, look for all of the inputs on the y-axis that correspond to points on the graph.

Example Use the graph of y = f(x) to solve this exercise. Domain and Range Graph Use the graph of y = f(x) to solve this exercise. What is f(4) – f(-3)? f(4) = 3, f(-3) = -2 f(4) – f(-3) = 5 What are the domain and range of f? Domain (-5, 6] Range [-4, 5] Example

Obtaining Information from Graphs To find the x-intercepts, look for the points at which the graph crosses the x- axis. The zeros of a function f are the x-values for which f(x) = 0. The real zeros are the x-intercepts. To find the y-intercept, look for the points at which the graph crosses the y-axis. A function can have more than one x- intercept but at most one y-intercept.

Example Use the graph of y = f(x) to solve this exercise. Recall that X- and y-Intercepts Graph Use the graph of y = f(x) to solve this exercise. What are the x-intercepts? -4, 1, 5 What is the y-intercept? -3 Recall that A function can have more than one x-intercept but at most one y-intercept. Example

Example Use the graph of y = f(x) to solve this exercise. Recall that Zeros Graph Use the graph of y = f(x) to solve this exercise. What are the zeros of f? -2, 2 Find the value(s) of x for which f(x) = -1. -1, 1 Find the value(s) of x for which f(x) = -2. Recall that the zeros of a function, f, are the x-values for which f(x) = 0. Example

Increasing, Decreasing, and Constant Functions A function is increasing on an open interval if the graph rises from left to right. A function is decreasing on an open interval if the graph falls from left to right. A function is constant on an open interval if it is horizontal (parallel to the x-axis).

Example Use the graph of y = f(x) to solve this exercise. Intervals Graph Use the graph of y = f(x) to solve this exercise. On which interval or intervals is f increasing? (-1,2) decreasing? (-5, -1) (2, 6) Example

Relative Maxima and Relative Minima A relative maximum occurs if the graph changes direction from rising to falling on a given interval. A relative minimum occurs if the graph changes direction from falling to rising on a given interval.

Example Use the graph of y = f(x) to solve this exercise. Relative Maximum Graph Use the graph of y = f(x) to solve this exercise. For what number does f have a relative maximum? 2, f(2) = 5 What is the relative maximum? (2, 5) Example

Example Use the graph of y = f(x) to solve this exercise. Relative Minimum Graph Use the graph of y = f(x) to solve this exercise. For what number does f have a relative minimum? -1, f(-1) = -4 What is the relative minimum? (-1, -4) Example

Even and Odd Functions The function f is even if f(-x) = f(x) for all x in the domain of f. The function f is odd if f(-x) = -f(x) for all x in the domain of f.

Example Let Find f(-x) Find –f(x) Neither since f(x) = x2 – x – 4 Even, odd, or neither Solution Let f(x) = x2 – x – 4 Find f(-x) f(-x) = (-x)2 –(-x) -4 f(-x) = x2 + x – 4 Find –f(x) -f(x) = -(x2 – x – 4) -f(x) = -x2 + x + 4 Neither since f(x) ≠ f(-x) and f(x) ≠ -f(x) Example

Symmetry The graph of an even function is symmetric with respect to the y-axis. The graph of an odd function is symmetric with respect to the origin.

Example Use the graph of y = f(x) to solve this exercise. Even, odd, or neither Graph Use the graph of y = f(x) to solve this exercise. Is f even, odd, or neither? even Remember that the function f is an even function if f(-x) = f(x). The graph is symmetric with respect to the y-axis. odd function if f(-x) = -f(x). The graph is symmetric with respect to the origin. Example

Graphs of Common Functions Knowing the graphs of common functions can help you analyze transformations into more complicated graphs.

Constant function

Identity function

Absolute value function

Standard quadratic function

Square root function

Standard cubic function

Cube root function

Transformations Vertical or horizontal shifts Reflections about the x- or y-axis Vertical stretching or shrinking Horizontal stretching or shrinking

Order of Transformations Horizontal shifting Stretching or shrinking Reflecting Vertical shifting

Example Graph g(x) = 4(x – 5)2 + 2 Problem Solution Graph g(x) = 4(x – 5)2 + 2 Start with the graph of the basic function f(x) = x2. Example

Example Graph g(x) = 4(x – 5)2 + 2 Look for horizontal shifting. Problem Solution Graph g(x) = 4(x – 5)2 + 2 Look for horizontal shifting. The x – 5 in parentheses tells us to shift the graph five units to the right. Example

Example Graph g(x) = 4(x – 5)2 + 2 Problem Solution Graph g(x) = 4(x – 5)2 + 2 Check for reflecting. This graph doesn’t require reflecting. Look for stretching or shrinking. The 4 in front of (x – 5)2 tells us to multiply each y- coordinate by 4, thus vertical stretching the graph. Example

Example Graph g(x) = 4(x – 5)2 + 2 Problem Solution Graph g(x) = 4(x – 5)2 + 2 Finally, check for vertical shifting. The + 2 tells us to raise the graph 2 units. Observe how the graphs transformed from the red to the gold one. Example

Combinations of Functions Functions can be Added Subtracted Multiplied Divided

Composition of Functions Another way of combining two functions. Denoted by Defined by

Combinations/Composite Functions Combinations of Functions Composite Functions (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) (fg)(x) = f(x)g(x) (f/g)(x) = f(x)/g(x) (f ∘ g)(x) = f(g(x)) (g ∘ f)(x) = g(f(x)) Combinations/Composite Functions

Example Let (x2 – x – 4) – (2x – 6) = x2 - x – 4 – 2x + 6 = (f – g)(x) (g – f)(x) Let f(x) = x2 – x – 4 g(x) = 2x – 6 (x2 – x – 4) – (2x – 6) = x2 - x – 4 – 2x + 6 = x2 - 3x + 2 Let f(x) = x2 – x – 4 g(x) = 2x – 6 (2x – 6) – (x2 – x – 4) 2x – 6 - x2 + x +4 = -x2 + 3x - 2 Example

Example Let Let f(x) = x2 – x – 4 g(x) = 2x – 6 f(x) = x2 – x – 4 (f ◦ g)(x) (g ◦ f)(x) Let f(x) = x2 – x – 4 g(x) = 2x – 6 Let f(x) = x2 – x – 4 g(x) = 2x – 6 Example

Example Let Let f(x) = 9x - 5 g(x) = (x + 9)/5 f(x) = 9x - 5 (f ◦ g)(x) or f(g(x)) (g ◦ f)(x) or g(f(x)) Let f(x) = 9x - 5 g(x) = (x + 9)/5 Let f(x) = 9x - 5 g(x) = (x + 9)/5 Example

Problem Solution The function g(x) = 3.9√x + 6.3 models the median height, g(x), in inches, of children who are x months of age. What values are represented by the x- and y-axes? x is age in months. y is median height. Using Modeling

Problem Solution The function g(x) = 3.9√x + 6.3 models the median height, g(x), in inches, of children who are x months of age. According to the model, what is the median height of children who are two years old? Evaluate the function for x = 24. 3.9√24 + 6.3 = 25.4 inches Using Modeling

Problem Solution The function g(x) = 3.9√x + 6.3 models the median height, g(x), in inches, of children who are x months of age. How well does the model describe the actual height? The graph is very close. Using Modeling

Problem Solution The function g(x) = 3.9√x + 6.3 models the median height, g(x), in inches, of children who are x months of age. Use the model to find the average rate of change in inches per month between birth and 12 months. Use the average rate of change of a function formula Using Modeling

Problem Solution The function g(x) = 3.9√x + 6.3 models the median height, g(x), in inches, of children who are x months of age. Use the model to find the average rate of change in inches per month between 30 and 42 months. Use the average rate of change of a function formula Using Modeling

Problem Solution The function g(x) = 3.9√x + 6.3 models the median height, g(x), in inches, of children who are x months of age. Compare the two average rates of change. How is this difference shown on the graph? The average rate of change is smaller. The graph is not as steep. Using Modeling

Inverse Functions If f(g(x)) = x and g(f(x)) = x, function g is the inverse of f. A one-to-one function has no two different ordered pairs with the same second component. Only one-to-one functions have inverse functions.

Horizontal Line Test A function f has an inverse if there is no horizontal line that intersects the graph of the function f at more than one point.

Example Use the graph of y = f(x) to solve this exercise. Inverse Functions Graph Use the graph of y = f(x) to solve this exercise. Does f have an inverse function? No Use the Horizontal Line Test to determine if there is an inverse. Example

Finding the Inverse of a Function Replace f(x) with y in the equation for f(x). Interchange x and y. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function. If f has an inverse function, replace y in Step 3 by f-1(x). Verify result by showing that f(f-1(x)) = x and f-1(f(x)) = x

Example Find the inverse of f(x) = 7x – 5 y = 7x – 5 x = 7y – 5 Solution Find the inverse of f(x) = 7x – 5 y = 7x – 5 x = 7y – 5 7y = x + 5, y = (x + 5)/7 f-1(x) = (x + 5)/7 Example