1. Show whether each function is one-to-one (injective). 1.i) Consider $f(x) = 10x + 7$. - A function is one-to-one if for any $x_1 \neq x_2$, we have $f(x_1) \neq f(x_2)$. - Here $f(x)$ is linear with slope 10, which is non-zero. - If $f(x_1) = f(x_2)$, then $10x_1 + 7 = 10x_2 + 7 \Rightarrow 10x_1 = 10x_2 \Rightarrow x_1 = x_2$. - So, $f(x)$ is one-to-one. 1.ii) Consider $f(x) = \sqrt{2x - 11}$. - The square root function is increasing and defined for $x \geq 5.5$. - Assume $f(x_1) = f(x_2) \Rightarrow \sqrt{2x_1 - 11} = \sqrt{2x_2 - 11} \Rightarrow 2x_1 - 11 = 2x_2 - 11 \Rightarrow x_1 = x_2$. - Hence, $f(x)$ is one-to-one on its domain. 1.iii) Consider $f(x) = -4x^2 + 1$. - This is a quadratic function opening downward (because coefficient of $x^2$ is negative). - For quadratic functions, generally $f(x_1) = f(x_2)$ occurs for $x_1 \neq x_2$ - For example: $f(1) = -4(1)^2 + 1 = -4 + 1 = -3$ and $f(-1) = -4(-1)^2 + 1 = -3$. - Since two different inputs map to the same output, this function is NOT one-to-one. 1.iv) Consider $f(x) = \sqrt[3]{3x + 5}$. - Cube root function is strictly increasing for all real numbers. - Assume $f(x_1) = f(x_2) \Rightarrow \sqrt[3]{3x_1 + 5} = \sqrt[3]{3x_2 + 5} \Rightarrow 3x_1 + 5 = 3x_2 + 5 \Rightarrow x_1 = x_2$. - Therefore, this function is one-to-one. 2. Determine the truth of statements about $f(x) = (x - 5)^2 + 5$ on given intervals. 2.i) On interval $[-1, 5]$. - $f(x)$ is a parabola opening upward with vertex at $x=5$. - On $[-1,5]$, the function is decreasing as $x$ approaches 5 from the left, so it is one-to-one decreasing on this interval. - More formally, the derivative $f'(x) = 2(x - 5)$ is negative for $x < 5$, so strictly decreasing on $[-1,5)$. - Hence, $f(x)$ is one-to-one on $[-1,5]$. 2.ii) On interval $[0,7]$. - $f(x)$ decreases from 0 to 5, then increases from 5 to 7. - So $f(x)$ is not one-to-one on $[0,7]$ because it takes the same value twice at points equidistant from 5. - For example, $f(3) = f(7)$. Thus, statement (i) is true and statement (ii) is false. 3. Check if the given sets of points represent one-to-one functions. 3.a) $f = \{(7,3), (8,-5), (-2,11), (-6,4)\}$ - Check if any $y$-value repeats. Outputs are $3, -5, 11, 4$ all distinct. - So function is one-to-one. 3.b) $f = \{(-3,8), (-11,-9), (5,4), (6,-9)\}$ - Outputs are $8, -9, 4, -9$. - The output $-9$ repeats for inputs $-11$ and $6$, so not one-to-one. Final answers: - 1.i one-to-one - 1.ii one-to-one - 1.iii not one-to-one - 1.iv one-to-one - 2.i true - 2.ii false - 3.a one-to-one - 3.b not one-to-one