If [tex]f(x)=\left\{\begin{array}{ll}2 x+1 & x\ \textless \ 0 \\ 2 x+2 & x \geq 0\end{array}\right.[/tex] Then find

1) [tex]f ( 0 )=2(0)+2=2[/tex]
2) [tex]f (- 1 )=2(-1)+1=-1[/tex]
3) [tex]f ( 2 )=2(2)+[/tex] ?
4) [tex]f ( 0 )+ 3 f (- 1 )- 5 f ( 2 )=2+3(-1)- S (6)=-31[/tex]
[tex]=6[/tex]
5) [tex]f\left(t^2+1\right)=[/tex]

Evaluate f ( 0 ) using the second case: f ( 0 ) = 2 ( 0 ) + 2 = 2 .
Evaluate f ( − 1 ) using the first case: f ( − 1 ) = 2 ( − 1 ) + 1 = − 1 .
Evaluate f ( 2 ) using the second case: f ( 2 ) = 2 ( 2 ) + 2 = 6 .
Calculate f ( 0 ) + 3 f ( − 1 ) − 5 f ( 2 ) = 2 + 3 ( − 1 ) − 5 ( 6 ) = − 31 .
Evaluate f ( t 2 + 1 ) using the second case: f ( t 2 + 1 ) = 2 ( t 2 + 1 ) + 2 = 2 t 2 + 4 . The final answer is 2 t 2 + 4 ​ .

Explanation

Understanding the Piecewise Function We are given a piecewise function: f ( x ) = { 2 x + 1 2 x + 2 ​ x < 0 x ≥ 0 ​ We need to evaluate the function at several points and also find an expression for f ( t 2 + 1 ) .

Evaluating the Function



To find f ( 0 ) , we use the second case since 0 ≥ 0 :
f ( 0 ) = 2 ( 0 ) + 2 = 2
To find f ( − 1 ) , we use the first case since − 1 < 0 :
f ( − 1 ) = 2 ( − 1 ) + 1 = − 2 + 1 = − 1
To find f ( 2 ) , we use the second case since 2 ≥ 0 :
f ( 2 ) = 2 ( 2 ) + 2 = 4 + 2 = 6
To find f ( 0 ) + 3 f ( − 1 ) − 5 f ( 2 ) , we substitute the values we found: f ( 0 ) + 3 f ( − 1 ) − 5 f ( 2 ) = 2 + 3 ( − 1 ) − 5 ( 6 ) = 2 − 3 − 30 = − 31
To find f ( t 2 + 1 ) , we note that t 2 ≥ 0 for any real number t . Therefore, 0"> t 2 + 1 ≥ 1 > 0 . We use the second case since t 2 + 1 ≥ 0 :
f ( t 2 + 1 ) = 2 ( t 2 + 1 ) + 2 = 2 t 2 + 2 + 2 = 2 t 2 + 4


Final Results Therefore,


f ( 0 ) = 2
f ( − 1 ) = − 1
f ( 2 ) = 6
f ( 0 ) + 3 f ( − 1 ) − 5 f ( 2 ) = − 31
f ( t 2 + 1 ) = 2 t 2 + 4

Examples
Piecewise functions are used in various real-world scenarios, such as defining tax brackets. For example, the amount of tax you pay might be different depending on your income level. If your income is below a certain threshold, you pay one tax rate, and if it's above that threshold, you pay a higher rate. This is a piecewise function where the tax rate changes based on income.

Answered by GinnyAnswer | 2025-07-05

To evaluate the piecewise function f ( x ) , we find f ( 0 ) = 2 , f ( − 1 ) = − 1 , and f ( 2 ) = 6 . Using these values, we calculate f ( 0 ) + 3 f ( − 1 ) − 5 f ( 2 ) = − 31 . For f ( t 2 + 1 ) , we find it equals 2 t 2 + 4 .
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Answered by Anonymous | 2025-07-08