Question

Find the domain and range of the function.

$f\left(x\right)=5x^2+4,0\le x\le 2$

Short answer

The domain of the given function is $\left[0,2\right]$and the range of the given function is $\left[4,24\right]$.

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:

$f\left(x\right)=5x^2+4,0\le x\le 2$

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 $0\le x\le 2$, so the domain is $\left[0,2\right]$.

It is given that,

$0\le x\le 2$

Square each side.

$\begin{array}{c}{0}^2\le x^2\le 2^2\\ 0\le x^2\le 4\end{array}$

Multiply each side by $5$.

$\begin{array}{c}5\left(0\right)\le 5\left(x^2\right)\le 5\left(4\right)\\ 0\le 5x^2\le 20\end{array}$

Add$4$ on each side.

$$\begin{gathered} 0+4\le 5x^2+4\le 20+4 \\ 4\le f\left(x\right)\le 24 \end{gathered}$$

Since $4\le f\left(x\right)\le 24$, therefore the range is $\left[4,24\right]$.

Thus, the domain of the given function is $\left[0,2\right]$and the range of the given function is $\left[4,24\right]$.