Example 2 Using the graphs below, evaluate . g(x) f(x) To evaluate , we again start with the inside evaluation. We evaluate using the graph of the function, finding the input of 1 on the horizontal axis and finding the output value of the g(x) graph at that input. Here, . Using this value as the input to the function, . We can then f evaluate this by looking to the graph of the function, finding the input of 3 on the f(x) horizontal axis, and reading the output value of the graph at this input. Here, , so. Composition Formulas When evaluating a composition of functions where we have either created or been given formulas, the concept of working from the inside out remains the same. First we evaluate the inside function using the input value provided, then use the resulting output as the input to the outside function. Example 3 Given and , evaluate . Since the inside evaluation iswe start by evaluating the function at 1: h(x) Then , so we evaluate the function at an input of 5: f(t) We are not limited, however, to using a numerical value as the input to the function. We can put anything into the function: a value, a different variable, or even an algebraic expression, provided we use the input expression everywhere we see the input variable. Example 4 Let and , find and f(g(x)) g(f(x)). To find we start by evaluating the inside, writing out the formula for f(g(x)), g(x) We then use the expression as input for the function f. We then evaluate the function using the formula for as the input. f(x) g(x) Since then This gives us the formula for the composition:
Example 2 Using the graphs below, evaluate . g(x) f(x) To evaluate , we again start with the inside evaluation. We evaluate using the graph of the g(x) function, finding the input of 1 on the horizontal axis and finding the output value of the graph at that input. Here, . Using this value as the input to the f function, . We can then evaluate this by looking to the graph of the f(x) function, finding the input of 3 on the horizontal axis, and reading the output value of the graph at this input. Here, , so. Composition Formulas When evaluating a composition of functions where we have either created or been given formulas, the concept of working from the inside out remains the same. First we evaluate the inside function using the input value provided, then use the resulting output as the input to the outside function. Example 3 Given and , evaluate . Since the inside evaluation iswe start by evaluating the h(x) function at 1: Then , so we evaluate the f(t) function at an input of 5: We are not limited, however, to using a numerical value as the input to the function. We can put anything into the function: a value, a different variable, or even an algebraic expression, provided we use the input expression everywhere we see the input variable. Example 4 Let and , find f(g(x)) and g(f(x)). To find f(g(x)), we start by evaluating the inside, writing out the formula for g(x) We then use the expression as input for the function f. We then evaluate the function f(x) using the formula for g(x) as the input. Since then This gives us the formula for the composition:
BusCalc_Chp1_wExercises.docx