What's [tex]f(g(4))[/tex] if [tex]f(x)=3 x^2-3 x+6[/tex] and [tex]g(x)=2 x[/tex]?
A) 12
B) 210
C) 174
D) 24
Answer (2)
Evaluate g ( 4 ) : g ( 4 ) = 2 × 4 = 8 .
Substitute g ( 4 ) into f ( x ) : f ( g ( 4 )) = f ( 8 ) = 3 × 8 2 − 3 × 8 + 6 .
Simplify the expression: f ( 8 ) = 192 − 24 + 6 = 174 .
The final answer is 174 .
Explanation
Understanding the Problem We are given two functions: f ( x ) = 3 x 2 − 3 x + 6 and g ( x ) = 2 x . Our goal is to find the value of the composite function f ( g ( 4 )) . This means we first need to evaluate g ( 4 ) , and then substitute that result into the function f ( x ) .
Evaluating g(4) First, let's find g ( 4 ) . We substitute x = 4 into the expression for g ( x ) : g ( 4 ) = 2 × 4 = 8
Substituting into f(x) Now that we have g ( 4 ) = 8 , we can find f ( g ( 4 )) by substituting 8 into the function f ( x ) : f ( g ( 4 )) = f ( 8 ) = 3 × 8 2 − 3 × 8 + 6
Simplifying the Expression Let's simplify the expression: f ( 8 ) = 3 × 64 − 3 × 8 + 6 = 192 − 24 + 6 = 174 So, f ( g ( 4 )) = 174 .
Final Answer Therefore, the value of f ( g ( 4 )) is 174. The correct answer is C) 174.
Examples
Composite functions are used in many real-world applications. For example, in manufacturing, if g ( x ) represents the number of products produced by x employees and f ( x ) represents the profit earned from x products, then f ( g ( x )) would represent the profit earned based on the number of employees. Evaluating composite functions helps businesses understand the relationship between different variables and optimize their operations.
To calculate f ( g ( 4 )) , we first find g ( 4 ) which equals 8. Then by substituting 8 into f ( x ) , we simplify to find that f ( g ( 4 )) = 174 . The correct answer is C) 174.
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