Full solution

Q. Given

f(x)=-x+3

and

g(x)=\frac{1}{2} x^{2}-1

, which is a solution to

f(x)=g(x)

\newline

-2

\newline

-1

\newline

1

\newline

2

Set Equation Equal: To find the solution to $f(x) = g(x)$ , we need to set the two functions equal to each other and solve for $x$ . So, we have the equation $-x + 3 = \left(\frac{1}{2}\right)x^2 - 1$ .
Form Quadratic Equation: First, let's move all terms to one side of the equation to set it to zero and form a quadratic equation. $\newline$ We add $x$ and subtract $3$ from both sides to get: $\newline$ $(\frac{1}{2})x^2 + x - 4 = 0$ .
Eliminate Fraction: To make the equation easier to solve, we can multiply every term by $2$ to get rid of the fraction: $\newline$ $2 \times (\frac{1}{2})x^2 + 2 \times x - 2 \times 4 = 0 \times 2$ , $\newline$ which simplifies to: $\newline$ $x^2 + 2x - 8 = 0$ .
Factor Quadratic Equation: Now, we need to factor the quadratic equation, if possible. $\newline$ The factors of $-8$ that add up to $2$ are $4$ and $-2$ . $\newline$ So, we can write the equation as: $\newline$ $(x + 4)(x - 2) = 0$ .
Apply Zero Product Property: Next, we apply the zero product property, which states that if a product of two factors is zero, then at least one of the factors must be zero. $\newline$ So, we set each factor equal to zero: $\newline$ $x + 4 = 0$ or $x - 2 = 0$ .
Solve for x: Solving each equation for x gives us the possible solutions: $x = -4$ or $x = 2$ .
Check Answer Choices: We need to check which of these solutions is given in the answer choices. $\newline$ The choices are A) $-2$ , B) $-1$ , C) $1$ , D) $2$ . $\newline$ Only $x = 2$ is in the answer choices, so $x = -4$ is not a valid option for this problem.
Verify Solution: Finally, we can verify that $x = 2$ is indeed a solution by substituting it back into the original equations $f(x)$ and $g(x)$ to see if they are equal. $\newline$ For $f(x)$ , we have $f(2) = -2 + 3 = 1$ . $\newline$ For $g(x)$ , we have $g(2) = (\frac{1}{2})(2)^2 - 1 = (\frac{1}{2})(4) - 1 = 2 - 1 = 1$ . $\newline$ Since $f(2) = g(2)$ , $x = 2$ is a solution to $f(x) = g(x)$ .