1. **Problem Statement:** Determine if each function $g: \mathbb{R} \to \mathbb{R}$ is one-to-one (injective) and onto (surjective). If not onto, find the range $g(\mathbb{R})$. 2. **Key Definitions:** - A function is **one-to-one** if different inputs produce different outputs: $g(x_1) = g(x_2) \implies x_1 = x_2$. - A function is **onto** if every real number is an output: for every $y \in \mathbb{R}$, there exists $x$ such that $g(x) = y$. --- ### a) $g(x) = x + 7$ 1. Check one-to-one: $$g(x_1) = g(x_2) \implies x_1 + 7 = x_2 + 7 \implies x_1 = x_2$$ So, $g$ is one-to-one. 2. Check onto: For any $y \in \mathbb{R}$, solve $y = x + 7 \implies x = y - 7$, which is real. So, $g$ is onto. --- ### b) $g(x) = 2x - 3$ 1. One-to-one: $$g(x_1) = g(x_2) \implies 2x_1 - 3 = 2x_2 - 3 \implies x_1 = x_2$$ So, one-to-one. 2. Onto: For any $y$, $y = 2x - 3 \implies x = \frac{y+3}{2}$ real. So, onto. --- ### c) $g(x) = -x + 5$ 1. One-to-one: $$g(x_1) = g(x_2) \implies -x_1 + 5 = -x_2 + 5 \implies x_1 = x_2$$ So, one-to-one. 2. Onto: For any $y$, $y = -x + 5 \implies x = 5 - y$ real. So, onto. --- ### d) $g(x) = x^2$ 1. One-to-one: Check if $g(x_1) = g(x_2)$ implies $x_1 = x_2$. But $g(1) = 1 = g(-1)$ and $1 \neq -1$, so not one-to-one. 2. Onto: Since $x^2 \geq 0$ for all real $x$, negative numbers are not in the range. Range: $g(\mathbb{R}) = [0, \infty)$. --- ### e) $g(x) = x^2 + x$ 1. One-to-one: Check if $g(x_1) = g(x_2)$ implies $x_1 = x_2$. Try $x=0$: $g(0)=0$, and $x=-1$: $g(-1)=1 -1=0$, so not one-to-one. 2. Onto: Rewrite as $g(x) = x^2 + x = x^2 + x + \frac{1}{4} - \frac{1}{4} = (x + \frac{1}{2})^2 - \frac{1}{4}$. Minimum value is $-\frac{1}{4}$ at $x = -\frac{1}{2}$. Range: $[-\frac{1}{4}, \infty)$. --- ### f) $g(x) = x^3$ 1. One-to-one: If $g(x_1) = g(x_2)$, then $x_1^3 = x_2^3 \implies x_1 = x_2$. So, one-to-one. 2. Onto: For any $y$, $y = x^3 \implies x = \sqrt[3]{y}$ real. So, onto. --- **Final answers:** - a) One-to-one: Yes, Onto: Yes - b) One-to-one: Yes, Onto: Yes - c) One-to-one: Yes, Onto: Yes - d) One-to-one: No, Onto: No, Range: $[0, \infty)$ - e) One-to-one: No, Onto: No, Range: $[-\frac{1}{4}, \infty)$ - f) One-to-one: Yes, Onto: Yes