182 Chapter 3 Graphs, Relations, and Functions In mathematics, we use letters such as f, g, and h to name functions. The symbol f (x) is read “f of x” and means “the function of x.” Function notation means that y is a function of x. In other words, y depends on the value of x. As a result, y is called the dependent variable and x is called the independent variable. Definition: The notation f (x) is function notation and denotes that y is a function of x. y = f (x) Output value Input value Name of function In this notation, x is the input value and f (x), or y, is the output value. We can also use function notation to make the process of finding input and output values easier. For instance, if we want to find the value of y = 5x + 2 when x = 3, we do the following. f (x) = 5x + 2 Write the equation with function notation. f (3) = 5(3) + 2 Replace x with 3. f (3) = 15 + 2 Multiply. f (3) = 17 Add. So, f (3) = 17 means that when x = 3, y = 17. It also means that the ordered pair (3, 17) is a solution of the equation y = 5x + 2. The process of finding an output value, y, for a given input value, x, is called evaluating a function. Procedure: Evaluating a Function f (x) at the Value k Step 1: If the function is given in terms of an equation, replace the variable with the given number k and simplify. Step 2: If the function is given in terms of a set, a mapping, or a table, find the y-value that corresponds to the x-value of k. Step 3: If the function is given in terms of a graph, find the ordered pair on the graph whose x-value is k. The corresponding y-value is f (k). We can also use function notation to solve an equation of the form f (x) = h, where h is a real number. For example, if f (x) = 5x + 2, to solve f (x) = 7 means to find the solution of 5x + 2 = 7. Solving this equation gives us x = 1. So, f (x) = 7 when x = 1. Procedure: Solving the Equation f (x) = h, Where h Is a Real Number Step 1: If the function is given in terms of an equation, replace f (x) with the value h and solve the resulting equation for x. Step 2: If the function is given in terms of a set, a mapping, or a table, find the x-value that corresponds to the y-value of h. Step 3: If the function is given in terms of a graph, find the ordered pair on the graph whose y-value is h. The corresponding x-value is the solution of f (x) = h. Note that there may be more than 1 solution of f (x) = h. Objective 3 Examples Find the requested information. 3a. Find g(-3) if g(x) = x2 + 1. 3b. Let f (x) = 3x - 12. Find f (0) and find x such that f (x) = 0.
hendricks_intermediate_algebra_1e_ch1_3
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