1. **Problem statement:** Given the function $f : \mathbb{R} \to [-1, \infty)$ defined by $f(x) = 4x^2 - 1$, find the domain and range of $f$ and sketch its graph. 2. **Domain:** The domain of a function is the set of all possible input values $x$ for which the function is defined. Since $f(x) = 4x^2 - 1$ is a polynomial, it is defined for all real numbers. \[ \text{Domain} = \mathbb{R} = (-\infty, \infty) \] 3. **Range:** To find the range, analyze the expression $4x^2 - 1$. - Since $x^2 \geq 0$ for all real $x$, the smallest value of $4x^2$ is 0. - Therefore, the smallest value of $f(x)$ is when $x=0$: $$ f(0) = 4 \times 0^2 - 1 = -1 $$ - As $x^2$ increases, $4x^2 - 1$ increases without bound. Hence, the range is all real numbers greater than or equal to $-1$: \[ \text{Range} = [-1, \infty) \] 4. **Graph sketch:** The function $f(x) = 4x^2 - 1$ is a parabola opening upwards with vertex at $(0, -1)$. - The vertex is the minimum point. - The parabola is symmetric about the $y$-axis. 5. **Summary:** - Domain: $(-\infty, \infty)$ - Range: $[-1, \infty)$ This completes the solution.