For the following function find (a) [tex]$f(4)$[/tex], (b) [tex]$f(-\frac{1}{2})$[/tex], (c) [tex]$f(a)$[/tex], (d) [tex]$f(\frac{2}{m})$[/tex], and (e) any values of [tex]$x$[/tex] such that [tex]$f(x)=1$[/tex].
[tex]$f(x)=5 x^2-45 x+101$[/tex]
(a) Find the value of [tex]$f(4)$[/tex].
[tex]$f(4)=\square$[/tex]
Answer (1)
Evaluate f ( 4 ) by substituting x = 4 into the function: f ( 4 ) = 5 ( 4 ) 2 − 45 ( 4 ) + 101 = 1 .
Evaluate f ( − 2 1 ) by substituting x = − 2 1 into the function: f ( − 2 1 ) = 5 ( − 2 1 ) 2 − 45 ( − 2 1 ) + 101 = 124.75 .
Evaluate f ( a ) by substituting x = a into the function: f ( a ) = 5 a 2 − 45 a + 101 .
Find x such that f ( x ) = 1 by solving the quadratic equation 5 x 2 − 45 x + 100 = 0 , which factors to ( x − 4 ) ( x − 5 ) = 0 , giving x = 4 and x = 5 . The final answers are: f ( 4 ) = 1 , f ( − 2 1 ) = 124.75 , f ( a ) = 5 a 2 − 45 a + 101 , f ( m 2 ) = m 2 20 − m 90 + 101 , and x = 4 , 5 .
Explanation
Problem Analysis We are given the function f ( x ) = 5 x 2 − 45 x + 101 and we need to find the values for (a) f ( 4 ) , (b) f ( − 2 1 ) , (c) f ( a ) , (d) f ( m 2 ) , and (e) the values of x such that f ( x ) = 1 .
Calculating f(4) (a) To find f ( 4 ) , we substitute x = 4 into the function: f ( 4 ) = 5 ( 4 ) 2 − 45 ( 4 ) + 101 = 5 ( 16 ) − 180 + 101 = 80 − 180 + 101 = 1
Calculating f(-1/2) (b) To find f ( − 2 1 ) , we substitute x = − 2 1 into the function: f ( − 2 1 ) = 5 ( − 2 1 ) 2 − 45 ( − 2 1 ) + 101 = 5 ( 4 1 ) + 2 45 + 101 = 4 5 + 4 90 + 4 404 = 4 499 = 124.75
Calculating f(a) (c) To find f ( a ) , we substitute x = a into the function: f ( a ) = 5 a 2 − 45 a + 101
Calculating f(2/m) (d) To find f ( m 2 ) , we substitute x = m 2 into the function: f ( m 2 ) = 5 ( m 2 ) 2 − 45 ( m 2 ) + 101 = 5 ( m 2 4 ) − m 90 + 101 = m 2 20 − m 90 + 101
Finding x such that f(x)=1 (e) To find the values of x such that f ( x ) = 1 , we set f ( x ) = 1 :
5 x 2 − 45 x + 101 = 1 5 x 2 − 45 x + 100 = 0 We can divide the equation by 5 to simplify it: x 2 − 9 x + 20 = 0 Now we can factor the quadratic equation: ( x − 4 ) ( x − 5 ) = 0 So the solutions are x = 4 and x = 5 .
Final Answer Therefore, the solutions are: (a) f ( 4 ) = 1 (b) f ( − 2 1 ) = 4 499 = 124.75 (c) f ( a ) = 5 a 2 − 45 a + 101 (d) f ( m 2 ) = m 2 20 − m 90 + 101 (e) x = 4 and x = 5
Examples
Understanding function evaluation is crucial in many real-world applications. For example, in physics, the height of a projectile over time can be modeled by a quadratic function. Evaluating the function at different times tells us the height of the projectile at those specific moments. Similarly, in economics, cost functions can be used to model the cost of producing a certain number of items. Evaluating the cost function helps businesses understand their expenses at different production levels, aiding in decision-making regarding pricing and production volume. This concept extends to various fields, including engineering, computer science, and finance, making it a fundamental skill in problem-solving and analysis.
