Question

Let \(f(x)=\begin{cases}1-5x,&\text{if } x<-2\\x^{2}-2x,&\text{if } -2\le x\le1\\-1+2x,&\text{if } x>1\end{cases}\). Then the value of \(f(-1)\) is equal to

• A. -3
• B. 3
• C. -1
• D. 1
• E. 0

Hint:

Algebra Tip: Always use parentheses when substituting negative numbers into an equation! Writing $(-1)^2$ ensures you get $+1$, whereas writing $-1^2$ evaluates to $-1$ and will lead to an incorrect answer.

Answer 1

The Correct Option is B

Concept:
A piecewise function defines different mathematical expressions based on the input value $x$. To evaluate $f(c)$, you must first determine which domain interval the constant $c$ falls into, and then substitute $c$ into the corresponding expression.

Step 1: Identify the input value.
We are asked to evaluate the function at $x = -1$.

Step 2: Determine the correct domain interval.
Look at the three given conditions for $x$: 1. $x < -2$ 2. $-2 \le x \le 1$ 3. $x > 1$ Since $-1$ is greater than $-2$ and less than $1$, it falls perfectly into the second interval ($-2 \le x \le 1$).

Step 3: Select the corresponding function expression.
The expression tied to the second interval is: $$f(x) = x^2 - 2x$$

Step 4: Substitute the input value into the expression.
Replace every instance of $x$ with $-1$: $$f(-1) = (-1)^2 - 2(-1)$$

Step 5: Calculate the final numerical result.
Perform the arithmetic: $$f(-1) = 1 - (-2)$$ $$f(-1) = 1 + 2 = 3$$ Hence the correct answer is (B) 3.