Function Evaluation Domain Ddf5E1

1. **Evaluate $f(2)$** Given $f(x)=2x^2-3x+1$, substitute $x=2$: $$f(2)=2(2)^2 - 3(2) + 1$$ Calculate powers and multiplication: $$= 2 \times 4 - 6 + 1$$ Simplify: $$= 8 - 6 + 1$$ $$= 3$$ 2. **Evaluate $f(-1)$** Substitute $x=-1$: $$f(-1)=2(-1)^2 - 3(-1) + 1$$ Calculate powers and multiplication: $$= 2 \times 1 + 3 + 1$$ Simplify: $$= 2 + 3 + 1$$ $$= 6$$ 3. **Evaluate $f(0)$** Substitute $x=0$: $$f(0)=2(0)^2 - 3(0) + 1$$ Calculate: $$= 0 - 0 + 1$$ $$= 1$$ 4. **Determine the domain of $f(x)=\frac{1}{x-5}$** The domain is all real numbers except where the denominator is zero. Set denominator equal to zero: $$x - 5 = 0$$ Solve for $x$: $$x = 5$$ So, domain is all real numbers except $x=5$. 5. **Determine the domain of $g(x)=2(x+8)$** This is a polynomial function, which is defined for all real numbers. Domain: all real numbers. 6. **Determine the domain of $h(x)=\frac{3}{x^2-16}$** Set denominator not equal to zero: $$x^2 - 16 \neq 0$$ Factor denominator: $$ (x-4)(x+4) \neq 0$$ So, $$x \neq 4$$ and $$x \neq -4$$ Domain: all real numbers except $x=4$ and $x=-4$. **Final answers:** 1. $f(2) = 3$ 2. $f(-1) = 6$ 3. $f(0) = 1$ 4. Domain of $f(x)=\frac{1}{x-5}$ is $\{x \in \mathbb{R} : x \neq 5\}$ 5. Domain of $g(x)=2(x+8)$ is all real numbers. 6. Domain of $h(x)=\frac{3}{x^2-16}$ is $\{x \in \mathbb{R} : x \neq 4, x \neq -4\}$