Chapter EP, Problem 7.3.3EP

To determine

To calculate: The domain and range of the function $y=\frac{1}{3}\sqrt{x+2}$ . Also sketch the graph of the function.

Answer to Problem 7.3.3EP

The domain is $\left\{x|x\ge -2\right\}$ range is $\left\{y|y\ge 0\right\}$ . Graph of the function is,

Explanation of Solution

Given information:

The square root function $y=\frac{1}{3}\sqrt{x+2}$ .

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 $y=\frac{1}{3}\sqrt{x+2}$ .

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,

  $y=\frac{1}{3}\sqrt{x-\left(-2\right)}+0$

The minimum point of the function is at $\left(h,k\right)$ that is $\left(-2,0\right)$ .

Next, construct the table of values for $x\ge -2$ .

  $\begin{array}{cc}x& y\\ -2& 0\\ -1& 0.3\\ 0& 0.47\\ 1& 0.57\\ 2& 0.66\end{array}$

Graph the coordinates obtained above.

Observe that the shape of the graph is same as that $y=\sqrt{x}$ but it is translated 2 units left and vertically compressed.

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

Next, the domain is $\left\{x|x\ge -2\right\}$ range is $\left\{y|y\ge 0\right\}$ .