Question

Domain and Range Find the domain and range of the function.

$f\left(x\right)=5x^2+4$

Short answer

The domain of the given function isall real numbers and the range of the given function is $\left[4,\infty \right)$.

Step-by-step solution

Step 1. Definition of domain and range.

The set of all input values of a function is known as the domain of the function.

The set of all output values of a function is known as the range of the function.

Step 2. Given information.

The given function is:

$f\left(x\right)=5x^2+4$

Step 3. Domain and range of the given function.

The given function is a quadratic function because the exponent of$x$ is $2$.

The given function is defined for all real values of $x$, so the domain is all real numbers.

It is clear that,

$x^2\ge 0$

Multiply both sides by $5$.

$5x^2\ge 0$

Add$4$ on both sides.

$\begin{array}{c}5x^2+4\ge 0+4\\ f\left(x\right)\ge 4\end{array}$

Since $f\left(x\right)\ge 4$, therefore the range is $\left[4,\infty \right)$.

Thus, the domain of the given function isall real numbers and the range of the given function is $\left[4,\infty \right)$.