1. **State the problem:** We are given two functions $f(x) = 2x^2 + 1$ and $g(x) = 2x - 6$. We need to determine which of the statements A through E about $f$ and $g$ are true. 2. **Recall definitions and formulas:** - The product of two functions $(f \cdot g)(x) = f(x) \times g(x)$. - The difference of two functions $(f - g)(x) = f(x) - g(x)$. - The sum of two functions $(f + g)(x) = f(x) + g(x)$. - The domain of $f + g$ is the intersection of the domains of $f$ and $g$. - The range depends on the function's outputs. 3. **Evaluate each statement:** **A. $f \cdot g$ is a quadratic function.** Calculate $f \cdot g$: $$ (f \cdot g)(x) = (2x^2 + 1)(2x - 6) = 2x^2 \times 2x - 2x^2 \times 6 + 1 \times 2x - 1 \times 6 = 4x^3 - 12x^2 + 2x - 6 $$ This is a cubic function (degree 3), not quadratic (degree 2). So, statement A is **false**. **B. $(f - g)(-2) = 19$** Calculate $(f - g)(-2)$: $$ (f - g)(-2) = f(-2) - g(-2) $$ Calculate $f(-2)$: $$ 2(-2)^2 + 1 = 2 \times 4 + 1 = 8 + 1 = 9 $$ Calculate $g(-2)$: $$ 2(-2) - 6 = -4 - 6 = -10 $$ So, $$ (f - g)(-2) = 9 - (-10) = 9 + 10 = 19 $$ Statement B is **true**. **C. The range of $f + g$ is $y \geq 5.5$.** Calculate $f + g$: $$ (f + g)(x) = (2x^2 + 1) + (2x - 6) = 2x^2 + 2x - 5 $$ This is a quadratic function with leading coefficient positive ($2$), so it opens upward. Vertex $x$-coordinate: $$ x = -\frac{b}{2a} = -\frac{2}{2 \times 2} = -\frac{2}{4} = -0.5 $$ Calculate vertex $y$-value: $$ (f + g)(-0.5) = 2(-0.5)^2 + 2(-0.5) - 5 = 2(0.25) - 1 - 5 = 0.5 - 1 - 5 = -5.5 $$ So the minimum value of $f + g$ is $-5.5$, not $5.5$. Statement C is **false**. **D. The domain of $f + g$ is all real numbers.** Both $f$ and $g$ are polynomials, which have domain all real numbers. Therefore, the domain of $f + g$ is all real numbers. Statement D is **true**. **E. The domain and range of $f \cdot g$ are all real numbers.** We found $f \cdot g$ is cubic: $4x^3 - 12x^2 + 2x - 6$. - Domain of a polynomial is all real numbers. - Range of a cubic polynomial is all real numbers (since cubic functions go to $\pm \infty$). So statement E is **true**. **Final answers:** - A: False - B: True - C: False - D: True - E: True **Summary:** Statements B, D, and E are true.