The functions $f$ and $g$ are defined as follows.
$f(x)=\frac{6 x+7}{5 x+3} \quad \text { and } \quad g(x)=\sqrt{x^2-8 x}$
Find $f\left(\frac{5}{x}\right)$ and $g(x-3)$.
Write your answers without parentheses and simplify them as much as possible.
$f\left(\frac{5}{x}\right)=\square$
$g(x-3)=\square
Answer (2)
The simplified expression for f ( x 5 ) is 25 + 3 x 30 + 7 x and for g ( x − 3 ) it is x 2 − 14 x + 33 .
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Substitute x 5 into f ( x ) and simplify: f ( x 5 ) = 5 ( x 5 ) + 3 6 ( x 5 ) + 7 = 25 + 3 x 30 + 7 x .
Substitute x − 3 into g ( x ) and simplify: g ( x − 3 ) = ( x − 3 ) 2 − 8 ( x − 3 ) = x 2 − 14 x + 33 .
The simplified expression for f ( x 5 ) is 25 + 3 x 30 + 7 x .
The simplified expression for g ( x − 3 ) is x 2 − 14 x + 33 .
f ( x 5 ) = 25 + 3 x 30 + 7 x g ( x − 3 ) = x 2 − 14 x + 33
Explanation
Problem Analysis We are given two functions, f ( x ) = 5 x + 3 6 x + 7 and g ( x ) = x 2 − 8 x . Our goal is to find f ( x 5 ) and g ( x − 3 ) , simplifying the expressions as much as possible and writing them without parentheses.
Finding f(5/x) First, let's find f ( x 5 ) . We substitute x 5 for x in the expression for f ( x ) : f ( x 5 ) = 5 ( x 5 ) + 3 6 ( x 5 ) + 7 To simplify this expression, we multiply both the numerator and the denominator by x :
f ( x 5 ) = x ( 5 ( x 5 ) + 3 ) x ( 6 ( x 5 ) + 7 ) = 5 ( 5 ) + 3 x 6 ( 5 ) + 7 x = 25 + 3 x 30 + 7 x So, f ( x 5 ) = 25 + 3 x 30 + 7 x .
Finding g(x-3) Next, let's find g ( x − 3 ) . We substitute x − 3 for x in the expression for g ( x ) : g ( x − 3 ) = ( x − 3 ) 2 − 8 ( x − 3 ) Now, we expand and simplify the expression inside the square root: ( x − 3 ) 2 = x 2 − 6 x + 9 8 ( x − 3 ) = 8 x − 24 So, we have: g ( x − 3 ) = ( x 2 − 6 x + 9 ) − ( 8 x − 24 ) = x 2 − 6 x + 9 − 8 x + 24 = x 2 − 14 x + 33 Thus, g ( x − 3 ) = x 2 − 14 x + 33 .
Final Answer Therefore, we have found that f ( x 5 ) = 25 + 3 x 30 + 7 x and g ( x − 3 ) = x 2 − 14 x + 33 .
Examples
Understanding function composition, as demonstrated in this problem, is crucial in many real-world applications. For instance, in physics, you might use composite functions to describe how the position of an object changes over time under the influence of a force field. Similarly, in economics, composite functions can model how production costs vary with the quantity of raw materials purchased, which in turn depends on market prices. By mastering these concepts, students can analyze and predict complex relationships in various fields, enhancing their problem-solving skills and analytical thinking.
