1. Problem: Find the value of the function $f(x) = 2x^2 - 3x + 1$ at $x=3$. 2. Formula: To find the value of a function at a specific point, substitute the value of $x$ into the function. 3. Calculation: $$f(3) = 2(3)^2 - 3(3) + 1 = 2(9) - 9 + 1 = 18 - 9 + 1 = 10$$ 4. Explanation: We replaced $x$ with 3 and simplified step-by-step to get the function value. 5. Final answer: $f(3) = 10$. --- 1. Problem: Determine if the function $g(x) = \frac{1}{x-2}$ is defined at $x=2$. 2. Formula: A function is undefined where the denominator is zero. 3. Calculation: Denominator at $x=2$ is $2-2=0$. 4. Explanation: Since division by zero is undefined, $g(2)$ does not exist. 5. Final answer: $g(2)$ is undefined. --- 1. Problem: Find the domain of the function $h(x) = \sqrt{5 - x}$. 2. Formula: The expression inside a square root must be non-negative. 3. Calculation: $$5 - x \geq 0 \Rightarrow x \leq 5$$ 4. Explanation: The domain includes all $x$ values less than or equal to 5. 5. Final answer: Domain is $(-\infty, 5]$. --- 1. Problem: Find the inverse of the function $f(x) = 3x + 4$. 2. Formula: To find the inverse, swap $x$ and $y$ and solve for $y$. 3. Calculation: $$y = 3x + 4 \Rightarrow x = 3y + 4 \Rightarrow 3y = x - 4 \Rightarrow y = \frac{x - 4}{3}$$ 4. Explanation: The inverse function is $f^{-1}(x) = \frac{x - 4}{3}$. 5. Final answer: $f^{-1}(x) = \frac{x - 4}{3}$. --- 1. Problem: Determine if the function $k(x) = x^2 - 4x + 4$ is one-to-one. 2. Formula: A function is one-to-one if each $y$ corresponds to exactly one $x$. 3. Calculation: Rewrite $k(x)$ as $(x-2)^2$. 4. Explanation: Since $(x-2)^2$ is a parabola opening upwards, it is not one-to-one because different $x$ values can give the same $y$. 5. Final answer: $k(x)$ is not one-to-one.