Chapter 0.2, Problem 7E

(a)

To determine

To find: The domain and range of the given function.

Answer to Problem 7E

The domain is equal to $\left[1,\infty \right)$ and the range is equal to $\left[2,\infty \right)$ .

Explanation of Solution

Given information:

The function is $y=2+\sqrt{x-1}$ .

Calculation:

The given function is $y=2+\sqrt{x-1}$ .

To find the domain: There does not exist a real square root of a negative number, so the inside of the square root, $x-1$ , must be greater than or equal to zero.

It can be simplified as below:

  $\begin{array}{c}x-1\ge 0\\ x\ge 1\end{array}$

So, the $y$ exists for all values of $x$ which are greater than or equal to $1$ .

Therefore, the domain is all real number greater than or equal to $1$ .

  $x\in \left[1,\infty \right)$

To find the range: The range of the function will be determined by the fact that the square root of a real number must always be positive.

The smallest value the function can take will happen for $x=1$ , since those values of $x$ will make the radical term equal to zero.

  $y=2+\sqrt{1-1}=2+0=2$ .

The range of the function will be $\left[2,\infty \right)$ .

Hence, the domain is $\left[1,\infty \right)$ and the range is $\left[2,\infty \right)$ .

(b)

To determine

To sketch: The graph of the function $y=2+\sqrt{x-1}$ .

Explanation of Solution

Given information:

The function is $y=2+\sqrt{x-1}$ .

Calculation:

The given function is $y=2+\sqrt{x-1}$ .

The given function is the parabolic function with vertex $\left(1,2\right)$ .

Now, sketch the graph of the function below: