Chapter EP, Problem 7.3.2EP

To determine

To calculate: The domain and range of the function $y=\sqrt{x+3}-1$ . Also sketch the graph of the function.

Answer to Problem 7.3.2EP

The domain is $\left\{x|x\ge -3\right\}$ range is $\left\{y|y\ge -1\right\}$ . Graph of the function is,

Explanation of Solution

Given information:

The square root function $y=\sqrt{x+3}-1$ .

Formula used:

The standard form of square root function is $f\left(x\right)=a\sqrt{x-h}+k$ . Here, h is the horizontal translation, k is the vertical translation and a is the orientation and shape with respect to the function $y=\sqrt{x}$ . The domain is $\left\{x|x\ge h\right\}$ and the range is $\left\{f\left(x\right)|f\left(x\right)\ge k\right\}$ .

Calculation:

Consider the square root function $y=\sqrt{x+3}-1$ .

Recall that the standard form of square root function is $f\left(x\right)=a\sqrt{x-h}+k$ . Here, h is the horizontal translation, k is the vertical translation and a is the orientation and shape with respect to the function $y=\sqrt{x}$ . The domain is $\left\{x|x\ge h\right\}$ and the range is $\left\{f\left(x\right)|f\left(x\right)\ge k\right\}$ .

Rewrite the function as,

  $y=1\cdot \sqrt{x-\left(-3\right)}+\left(-1\right)$

The minimum point of the function is at $\left(h,k\right)$ that is $\left(-3,-1\right)$ .

Next, construct the table of values for $x\ge -3$ .

  $\begin{array}{cc}x& y\\ -3& -1\\ -2& 0\\ -1& 0.41\\ 0& 0.73\\ 1& 1\end{array}$

Graph the coordinates obtained above.

Observe that the shape of the graph is same as that $y=\sqrt{x}$ but it is translated 3 units left and 1 unit down.

Also, as the value of x increases the corresponding values of y increases.

Next, the domain is $\left\{x|x\ge -3\right\}$ range is $\left\{y|y\ge -1\right\}$ .