$f(x)=\left\{\begin{array}{ll} |x|+1 & x<1 \\ -x+3 & x \geq 1 \end{array}\right.
f(2)=-2+3=1
a) $f(0)=|0|+1=1
b) $f(-3)=|-3|+1=4
c) $f(1)=$ $-(1)+3=2
d) $f(3)=-(3)+3=0
E) $f(0)-3 f(-3)+f(2)=1-3(4)+a=-10
f) $f\left(b^2+1\right)=-\left(b^2+1\right)^{-}=-b^2-1
g) The value of $f\left(-b^2-1\right)$ ?
A: $b^2+1$
B: $b^2+2$
C:$-b^2$
D: $b^2+2
Answer (2)
Determine that since − b 2 − 1 < 1 , the first case of the piecewise function applies: f ( x ) = ∣ x ∣ + 1 .
Substitute x = − b 2 − 1 into the first case: f ( − b 2 − 1 ) = ∣ − b 2 − 1∣ + 1 .
Simplify the absolute value: ∣ − b 2 − 1∣ = b 2 + 1 .
Calculate the final result: f ( − b 2 − 1 ) = b 2 + 1 + 1 = b 2 + 2 . The answer is b 2 + 2 .
Explanation
Determine which case to use We are given a piecewise function f ( x ) and asked to find the value of f ( − b 2 − 1 ) . First, we need to determine which case of the piecewise function to use.
Apply the correct case The function is defined as: f ( x ) = { ∣ x ∣ + 1 − x + 3 x < 1 x ≥ 1 We need to evaluate f ( − b 2 − 1 ) . Since b 2 is always non-negative, b 2 ≥ 0 . Therefore, − b 2 ≤ 0 , and − b 2 − 1 ≤ − 1 . Since − 1 < 1 , we use the first case of the piecewise function, which is f ( x ) = ∣ x ∣ + 1 .
Evaluate the absolute value Now we substitute x = − b 2 − 1 into f ( x ) = ∣ x ∣ + 1 : f ( − b 2 − 1 ) = ∣ − b 2 − 1∣ + 1 Since b 2 ≥ 0 , we have − b 2 − 1 < 0 . Therefore, ∣ − b 2 − 1∣ = − ( − b 2 − 1 ) = b 2 + 1 .
Calculate the final result So, f ( − b 2 − 1 ) = b 2 + 1 + 1 = b 2 + 2 .
State the final answer The value of f ( − b 2 − 1 ) is b 2 + 2 . The correct answer is B.
Examples
Understanding piecewise functions is crucial in many real-world applications. For instance, consider a cell phone plan where you pay a fixed rate for a certain amount of data, and then a different rate for any data you use beyond that limit. This is a piecewise function! Similarly, tax brackets work in a piecewise manner, where different income ranges are taxed at different rates. By understanding how to evaluate these functions, you can make informed decisions about your cell phone usage, understand your tax obligations, and more.
The value of f ( − b 2 − 1 ) using the provided piecewise function evaluates to b 2 + 2 . Therefore, the correct answer is option D. This conclusion is drawn by determining which part of the piecewise function to apply based on the value of the expression.
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