Problem 22

Question

Graph each of the functions. $$f(x)=2 \sqrt{x}$$

Step-by-Step Solution

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Answer

Graph the function using points (0, 0), (1, 2), and (4, 4). The domain is $x \geq 0$, and it's a vertical stretch of the square root function.

1Step 1: Understanding the Function

The function given is $ f(x) = 2 \sqrt{x} $. This is a square root function multiplied by a factor of 2. The square root function $ \sqrt{x} $ is defined for $ x \geq 0 $, so the domain of this function is also $ x \geq 0 $. The factor of 2 will vertically stretch the graph of the basic square root function.

2Step 2: Identifying Key Points

To create the graph, let's find some key points by plugging in some values of $ x $ into the function. Example points include: $ (0, 0) $ because $ f(0) = 2\sqrt{0} = 0 $, $ (1, 2) $ because $ f(1) = 2\sqrt{1} = 2 $, $ (4, 4) $ because $ f(4) = 2\sqrt{4} = 4 $.

Key Concepts

Key points for graphingFunction domainVertical stretching

Key points for graphing

When graphing the function $ f(x) = 2 \sqrt{x} $, identifying key points is essential, as it helps shape the curve. You can start by determining points where $ x $ is easy to compute with. Plugging these into the function gives you coordinates to plot.

Start with $ x = 0 $: $ f(0) = 2 \times \sqrt{0} = 0 $, giving you the point $ (0, 0) $.
Then let $ x = 1 $: $ f(1) = 2 \times \sqrt{1} = 2 $, leading to the point $ (1, 2) $.
Finally, try $ x = 4 $: $ f(4) = 2 \times \sqrt{4} = 4 $, resulting in the point $ (4, 4) $.

These points illustrate the rising nature of the curve. As $ x $ increases, the value of $ f(x) $ also rises, following the arc of a square root graph. Always start plotting from $ (0, 0) $ since this is the lowest point of the square root function.

Function domain

The domain of a function is critical as it defines the set of all possible input values. For square root functions, such as $ f(x) = 2 \sqrt{x} $, the domain is determined by the radicand, which in this case is $ x $.
For $ \sqrt{x} $ to be real, $ x $ must be non-negative. Hence, the function is only defined when:

$ x \geq 0 $

This means you start evaluating and graphing the function from $ x = 0 $ and continue forward. If any negative number is used as $ x $, the function would not output a real number, meaning it's out of the domain. This restriction explains why the function graph begins at the origin and moves right.

Vertical stretching

Vertical stretching occurs when the output of a function, or $ f(x) $, is multiplied by a factor. In $ f(x) = 2 \sqrt{x} $, the factor is 2, which causes vertical stretching.
This means that every output value of the basic square root function $ \sqrt{x} $ is doubled.

Without stretching: if $ x = 1 $, then $ f(x) = \sqrt{1} = 1 $.
With stretching: $ f(x) = 2 \times \sqrt{1} = 2 $.

By vertically stretching, the graph appears to extend taller compared to the basic function $ \sqrt{x} $. Each point on the graph rises higher, but the overall shape remains unchanged. This means that while the curve is stretched upward, it does not affect the domain or the direction in which the curve rises. Vertical stretching facilitates higher growth rates for the graph without altering its fundamental path.

Problem 22

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