Domain Range B672Eb

1. Find domain and range of $f(x) = 3x - 5$. - Domain: Since $f(x)$ is a linear function, it is defined for all real numbers. $$\text{Domain} = (-\infty, \infty)$$ - Range: Linear functions also have all real numbers as range. $$\text{Range} = (-\infty, \infty)$$ 2. Find domain and range of $y = -x^2 - 3$. - Domain: Polynomial functions are defined for all real $x$. $$\text{Domain} = (-\infty, \infty)$$ - Range: Since $-x^2$ is always negative or zero, the maximum value is at $x=0$: $$y = -0^2 - 3 = -3$$ The parabola opens downward, so range is: $$\text{Range} = (-\infty, -3]$$ 3. Find domain and range of $g(x) = -2\sqrt{x + 1}$. - Domain: The expression under the square root must be non-negative: $$x + 1 \geq 0 \Rightarrow x \geq -1$$ $$\text{Domain} = [-1, \infty)$$ - Range: Since $\sqrt{x+1} \geq 0$, multiplying by $-2$ makes it non-positive: $$g(x) \leq 0$$ Minimum value at $x=-1$: $$g(-1) = -2\sqrt{0} = 0$$ As $x \to \infty$, $g(x) \to -\infty$. $$\text{Range} = (-\infty, 0]$$ 4. Find domain and range of $h(x) = \sqrt{1 - x}$. - Domain: Inside the root must be non-negative: $$1 - x \geq 0 \Rightarrow x \leq 1$$ $$\text{Domain} = (-\infty, 1]$$ - Range: Square root outputs non-negative values. Minimum at $x=1$: $$h(1) = 0$$ Maximum at $x \to -\infty$ is unbounded but since $1-x$ grows large, $$\text{Range} = [0, \infty)$$ 5. Find domain and range of $y = \sqrt{3 + x} - 2$. - Domain: Inside root non-negative: $$3 + x \geq 0 \Rightarrow x \geq -3$$ $$\text{Domain} = [-3, \infty)$$ - Range: $\sqrt{3+x} \geq 0$, so minimum value of $y$ is at $x=-3$: $$y(-3) = 0 - 2 = -2$$ As $x \to \infty$, $y \to \infty$. $$\text{Range} = [-2, \infty)$$ 6. Find domain and range of $G(x) = -|x| + 5$. - Domain: Absolute value defined for all real $x$. $$\text{Domain} = (-\infty, \infty)$$ - Range: $|x| \geq 0$, so $- |x| \leq 0$, maximum at $x=0$: $$G(0) = 5$$ As $|x| \to \infty$, $G(x) \to -\infty$. $$\text{Range} = (-\infty, 5]$$ 7. Find domain and range of $y = \frac{1 - x}{2x}$. - Domain: Denominator $2x \neq 0 \Rightarrow x \neq 0$. $$\text{Domain} = (-\infty, 0) \cup (0, \infty)$$ - Range: Solve for $x$ in terms of $y$: $$y = \frac{1 - x}{2x} \Rightarrow 2xy = 1 - x$$ $$2xy + x = 1$$ $$x(2y + 1) = 1$$ $$x = \frac{1}{2y + 1}$$ - $x$ undefined if denominator zero: $$2y + 1 \neq 0 \Rightarrow y \neq -\frac{1}{2}$$ - So range is all real numbers except $-\frac{1}{2}$: $$\text{Range} = (-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)$$