Evaluate the piecewise function at the given values of the independent variable.

[tex]h(x)=\left\{\begin{array}{ll}\frac{x^2-4}{x-2} & \text { if } x \neq 2 \\4 & \text { if } x=2\end{array}\right.[/tex]

(a) [tex]h (3)[/tex]
(b) [tex]h(0)[/tex]
(c) [tex]h (2)[/tex]

Evaluate h ( 3 ) using the first part of the piecewise function since 3  = 2 : h ( 3 ) = 3 − 2 3 2 − 4 ​ = 5 .
Evaluate h ( 0 ) using the first part of the piecewise function since 0  = 2 : h ( 0 ) = 0 − 2 0 2 − 4 ​ = 2 .
Evaluate h ( 2 ) using the second part of the piecewise function since x = 2 : h ( 2 ) = 4 .
The final answers are: h ( 3 ) = 5 ​ , h ( 0 ) = 2 ​ , and h ( 2 ) = 4 ​ .

Explanation

Understanding the Problem We are given a piecewise function h ( x ) and asked to evaluate it at three different values of x : x = 3 , x = 0 , and x = 2 . The function is defined as follows:

h ( x ) = { x − 2 x 2 − 4 ​ 4 ​ if x  = 2 if x = 2 ​
We need to determine which part of the definition applies for each value of x and then evaluate the function accordingly.

Calculating h(3) (a) To find h ( 3 ) , we note that 3  = 2 , so we use the first part of the definition:

h ( 3 ) = 3 − 2 3 2 − 4 ​ = 1 9 − 4 ​ = 1 5 ​ = 5 .
Therefore, h ( 3 ) = 5 .

Calculating h(0) (b) To find h ( 0 ) , we note that 0  = 2 , so we use the first part of the definition:

h ( 0 ) = 0 − 2 0 2 − 4 ​ = − 2 0 − 4 ​ = − 2 − 4 ​ = 2 .
Therefore, h ( 0 ) = 2 .

Calculating h(2) (c) To find h ( 2 ) , we note that x = 2 , so we use the second part of the definition:

h ( 2 ) = 4 .
Therefore, h ( 2 ) = 4 .

Final Answer In summary:

(a) h ( 3 ) = 5 (b) h ( 0 ) = 2 (c) h ( 2 ) = 4
Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input. For example, consider a cell phone plan where you pay a fixed monthly fee for a certain amount of data, and then you pay an additional fee for each gigabyte of data you use beyond that limit. This can be modeled as a piecewise function, where the cost is constant up to the data limit and then increases linearly with additional data usage. Another example is income tax brackets, where the tax rate changes based on income levels, resulting in a piecewise function that determines the amount of tax owed.

Answered by GinnyAnswer | 2025-07-03

The evaluations of the function are: h ( 3 ) = 5 , h ( 0 ) = 2 , and h ( 2 ) = 4 .
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Answered by Anonymous | 2025-07-04