Determine the value of $f(3)$ for the function
$f(x)=\left\{\begin{array}{cc}\frac{x^3+18 x^2+104 x+192}{3 x-9} & \text { for } x \leq-3 \\ \frac{3 x^2-5 x+6}{x^3-2 x^2-9} & \text { for }-33\end{array}\right..$
Answer (2)
Determine which piece of the piecewise function to use based on the value of x .
Substitute x = 3 into the appropriate expression: f ( 3 ) = ( 3 ) 3 − 2 ( 3 ) 2 − 9 3 ( 3 ) 2 − 5 ( 3 ) + 6 .
Simplify the numerator and denominator: Numerator = 18, Denominator = 0.
Since the denominator is 0, the function is undefined: Undefined .
Explanation
Understanding the Problem We are given a piecewise function f ( x ) and asked to find the value of f ( 3 ) . The function is defined as: f ( x ) = { 3 x − 9 x 3 + 18 x 2 + 104 x + 192 x 3 − 2 x 2 − 9 3 x 2 − 5 x + 6 for x ≤ − 3 for − 33
Choosing the Correct Piece To find f ( 3 ) , we need to determine which piece of the function applies when x = 3 . Since − 3 < x ≤ 3 , we use the second piece of the function: f ( x ) = x 3 − 2 x 2 − 9 3 x 2 − 5 x + 6
Substituting x=3 Now, we substitute x = 3 into the expression: f ( 3 ) = ( 3 ) 3 − 2 ( 3 ) 2 − 9 3 ( 3 ) 2 − 5 ( 3 ) + 6
Simplifying the Expression Next, we simplify the numerator and the denominator separately. The numerator is: 3 ( 3 ) 2 − 5 ( 3 ) + 6 = 3 ( 9 ) − 15 + 6 = 27 − 15 + 6 = 12 + 6 = 18 The denominator is: ( 3 ) 3 − 2 ( 3 ) 2 − 9 = 27 − 2 ( 9 ) − 9 = 27 − 18 − 9 = 9 − 9 = 0
Final Answer So, we have: f ( 3 ) = 0 18 Since division by zero is undefined, f ( 3 ) is undefined.
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, a cell phone plan might charge a certain rate for the first 100 minutes of calls and a different rate for each additional minute. Similarly, income tax brackets are defined using a piecewise function, where the tax rate changes as income increases. Understanding how to evaluate piecewise functions is essential for analyzing and predicting outcomes in these scenarios.
To find f ( 3 ) , we use the second piece of the piecewise function since -3"> 3 > − 3 . After substituting, we find that the numerator equals 18, but the denominator equals 0, making f ( 3 ) undefined. Therefore, f ( 3 ) is undefined due to division by zero.
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