An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Find the composite function: g ( f ( x )) = f ( x ) − 1 = − 2 x − 5 − 1 = − 2 x − 6 .
Determine two points on the line: ( 0 , − 6 ) and ( − 3 , 0 ) .
Plot these points on the graph.
Draw a line through the points to represent the function g ( f ( x )) = − 2 x − 6 . The final answer is the graph of the line. g ( f ( x )) = − 2 x − 6
Explanation
Understanding the Problem We are given two functions, f ( x ) = − 2 x − 5 and g ( x ) = x − 1 . Our goal is to find the composite function g ( f ( x )) and then graph it.
Finding the Composite Function To find the composite function g ( f ( x )) , we need to substitute f ( x ) into g ( x ) . This means we replace the x in g ( x ) with the entire expression for f ( x ) . So, g ( f ( x )) = f ( x ) − 1 .
Substituting f(x) Now, we substitute f ( x ) = − 2 x − 5 into the expression for g ( f ( x )) : g ( f ( x )) = ( − 2 x − 5 ) − 1
Simplifying the Expression Next, we simplify the expression: g ( f ( x )) = − 2 x − 5 − 1 = − 2 x − 6 So, the composite function is g ( f ( x )) = − 2 x − 6 .
Finding Two Points on the Line The composite function g ( f ( x )) = − 2 x − 6 is a linear function. To graph this line, we need to find two points on the line. Let's choose x = 0 and x = − 3 .
When x = 0 :
y = − 2 ( 0 ) − 6 = − 6 So, the point is ( 0 , − 6 ) .
When x = − 3 :
y = − 2 ( − 3 ) − 6 = 6 − 6 = 0 So, the point is ( − 3 , 0 ) .
Graphing the Line Now we plot the two points ( 0 , − 6 ) and ( − 3 , 0 ) on the graph and draw a line through them. The line represents the composite function g ( f ( x )) = − 2 x − 6 .
Examples
Understanding composite functions is crucial in many real-world applications. For instance, consider a store offering a discount f ( x ) on an item, followed by a sales tax g ( x ) on the discounted price. The composite function g ( f ( x )) calculates the final price you pay after both the discount and tax are applied. This concept is also used in physics to analyze motion under multiple forces, and in computer graphics to combine transformations like rotations and scaling.
