Question

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the graph of the function $f(x) = -\frac{1}{2}x^2 + 2x - 1$ is shown.

graph of a parabola opening downward with vertex at (2, 1), passing through (0, -1) and (4, -1)

what is the domain of the function?

a $-3 < x < 7$

b $-10 \leq y \leq 1$

c all real numbers

Explanation:

Step1: Recall Domain Definition

Domain of a function is the set of all possible \( x \)-values (input values) for which the function is defined. For a quadratic function (polynomial function of degree 2), polynomials are defined for all real numbers because there are no restrictions like division by zero or square roots of negative numbers. The given function is \( f(x)=-\frac{1}{2}x^{2}+2x - 1 \), which is a quadratic (polynomial) function.

Step2: Analyze the Graph

Looking at the graph of the parabola, the arrows on the ends of the parabola extend infinitely to the left and right along the \( x \)-axis, indicating that there are no restrictions on the \( x \)-values. So, the \( x \)-values (domain) can be any real number.

Step3: Evaluate Options

• Option A: \( - 3
• Option B: \( - 10\leq y\leq1 \) is about \( y \)-values (range), not domain.
• Option C: "all real numbers" matches the definition of the domain for a quadratic function and the graph's behavior.

Answer:

C. all real numbers