If [tex]f(x)=3-2 x[/tex] and [tex]g(x)=\frac{1}{x+5}[/tex], what is the value of [tex]\left(\frac{f}{g}\right)(8) ?[/tex]
Answer (1)
First, find the value of f ( 8 ) by substituting x = 8 into f ( x ) = 3 − 2 x , which gives f ( 8 ) = − 13 .
Next, find the value of g ( 8 ) by substituting x = 8 into g ( x ) = x + 5 1 , which gives g ( 8 ) = 13 1 .
Then, calculate ( g f ) ( 8 ) = g ( 8 ) f ( 8 ) = 13 1 − 13 .
Finally, simplify the expression to get ( g f ) ( 8 ) = − 169 , so the answer is − 169 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 3 − 2 x and g ( x ) = x + 5 1 . We need to find the value of ( g f ) ( 8 ) , which means we need to calculate g ( 8 ) f ( 8 ) .
Calculating f(8) First, let's find the value of f ( 8 ) . We substitute x = 8 into the function f ( x ) : f ( 8 ) = 3 − 2 ( 8 ) = 3 − 16 = − 13
Calculating g(8) Next, let's find the value of g ( 8 ) . We substitute x = 8 into the function g ( x ) : g ( 8 ) = 8 + 5 1 = 13 1
Calculating (f/g)(8) Now, we can find the value of ( g f ) ( 8 ) by dividing f ( 8 ) by g ( 8 ) : ( g f ) ( 8 ) = g ( 8 ) f ( 8 ) = 13 1 − 13 = − 13 × 13 = − 169
Final Answer Therefore, the value of ( g f ) ( 8 ) is − 169 .
Examples
Understanding function composition and evaluation is crucial in many real-world applications. For example, in physics, you might use one function to describe the position of an object over time and another function to describe the force acting on that object. Combining these functions allows you to analyze how the force affects the object's position. Similarly, in economics, you could use functions to model supply and demand, and then combine them to determine the equilibrium price and quantity in a market. Evaluating these combined functions at specific points helps in making informed decisions and predictions.
