Question: 1. Find the domain and range of the following functions.
(a) f(x) = 2 - 3x
(b) g(x) = 3x² - 1
(c) f(x) = (x - 2)² + 5
(d) g(x) = x² + 4x - 1
(e) f(x) = \frac{2}{x+2}
(f) g(x) = \frac{x-1}{x+5}
(g) f(x) = \sqrt{x + 10}
(h) g(x) = |x - 3|
(i) f(x) = e^{-2x} + 5
(j) g(x) = \ln \sqrt{x + 2}
1. **Problem:** Find the domain and range of the function $$g(x) = 3x^2 - 1$$.
2. **Domain:** Since $$g(x)$$ is a polynomial, it is defined for all real numbers.
$$\text{Domain} = (-\infty, \infty)$$
3. **Range:** The function is a quadratic with positive leading coefficient (3), so it opens upwards.
The vertex is at $$x=0$$ (since no linear term), and
$$g(0) = 3(0)^2 - 1 = -1$$
Since it opens upwards, the minimum value is $$-1$$.
$$\text{Range} = [-1, \infty)$$
4. **Summary:**
- Domain: $$(-\infty, \infty)$$
- Range: $$[-1, \infty)$$