Chapter 5.7, Problem 13E

Solution

To compare: the properties of two functions and the key characteristics of their graph.

The domain and the range of the first function is $x\ge 0$ and $y\ge 1$ .

The domain and the range of the second function is $x\ge 1$ and $y\ge 5$ .

Neither of the graphs has asymptotes since graphs approach to $+\infty$ .

The second graph increases faster than the first graph.

Given information:

Function 1: a square root function whose graph passes through the points $\left(0,0\right),\left(1,1\right),\left(4,2\right),\left(9,3\right)$

Function 2: $y=3\sqrt{x-1}+5$

Concept Used:

Domain: The domain of a function is the set of all possible inputs $x$ for the function.

Range: The range of a function is the set of all possible outputs $y$ .

Asymptote: A vertical asymptote is a vertical line at which the function is undefined. The graph of a function cannot touch a vertical asymptote, and instead, the graph either goes up or down infinitely. A horizontal asymptote is a horizontal line that the graph of the function approaches as $x$ increases in the positive or negative direction. Unlike vertical asymptotes, it is possible to have the graph of a function touch its horizontal asymptote.

End Behaviour: The end behavior of a polynomial function is the behavior of the graph of as $x$ approaches positive infinity or negative infinity.

Calculation:

Function 1: a square root function whose graph passes through the points $\left(0,0\right),\left(1,1\right),\left(4,2\right),\left(9,3\right)$

Function 2: $y=3\sqrt{x-1}+5$

The graph is shown below:

  

Domain of the first function: $x\ge 0$

Range of the first function: $y\ge 1$

Domain of the second function: $x\ge 1$

Range of the second function: $y\ge 5$

Both graphs doesn’t have asymptote since graph approaches to $+\infty$ .

The second graph increases faster than the first graph.