Question

Find the domain and range of the function. $$ g(x)=x^{2}-5 $$

Short answer

The domain of the function \( g(x)=x^{2}-5 \) is \( (-\infty, +\infty) \) and the range is \( [-5, +\infty) \).

Step-by-step solution

Determine the domain

As this is a quadratic function, it is defined for all real numbers. Therefore, the domain of the function \( g(x) \) is \( (-\infty, +\infty) \).

Determine the vertex of the parabola

The vertex form of a quadratic function is \( h(x)=a(x-h)^{2}+k \) where (h, k) is the vertex of the parabola. This function \( g(x) \) is already in vertex form, \( g(x) = (x-0)^{2}-5 \), the vertex of the parabola is (0, -5).

Determine the direction of the parabola

The coefficient of \( x^{2} \) in the function \( g(x) = x^{2}-5 \) is 1, which is greater than 0. Therefore, the parabola opens upwards.

Determine the range

Since the parabola opens upwards and the vertex is (0, -5), the minimum value of the function \( g(x) \) is -5. Therefore, the range of the function \( g(x) \) is \( [-5, +\infty) \)