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Chapter 1: Problem 18

Find the natural domain and graph the functions. \(g(x)=\sqrt{-x}\)

Short Answer

Expert verified
The domain is \((-\infty, 0]\), and the graph is a left-opening half-parabola starting at the origin.

Step by step solution

01

Understand the Function

The function given is \( g(x) = \sqrt{-x} \). It involves a square root, which implies that the expression inside the root should be greater than or equal to zero for real number outputs.
02

Determine the Domain

To find the domain, identify the values of \( x \) for which \( -x \geq 0 \). This inequality simplifies to \( x \leq 0 \), meaning \( g(x) \) is defined for all \( x \leq 0 \). Thus, the natural domain of the function is \((-\infty, 0]\).
03

Graph the Function

To graph \( g(x) = \sqrt{-x} \), consider that it is the reflection of the graph of \( f(x) = \sqrt{x} \) across the y-axis. Plot points like \( x = -1 \), \( x = -4 \), etc., and connect them smoothly, noting that the vertex starts at the origin \((0,0)\). The graph is a half parabola opening to the left.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Function
The square root function is a fundamental concept in mathematics. It involves finding a number which, when multiplied by itself, gives the original number under the square root. For example, in the function \( g(x) = \sqrt{-x} \), the expression inside the square root can only be non-negative, because the square root of a negative number is not real. This leads us to our next topic, which is crucial in determining when the square root function is defined.
Inequalities
Inequalities are mathematical expressions involving symbols such as \(<, \leq, >, \geq\). They are used to describe a range of possible values. For the function \( g(x) = \sqrt{-x} \), we need to determine when \(-x\) is non-negative. This involves solving the inequality \(-x \geq 0\), which simplifies to \(x \leq 0\). This means that the function is defined only when \(x\) is less than or equal to zero. Here, inequalities help us establish the domain of the function, where it can be graphically represented with real outputs.
Graphing Functions
Graphing functions is a key method to visualize mathematical concepts. To graph \( g(x) = \sqrt{-x} \), you start by identifying the points that satisfy the function's domain鈥攊n this case, \(x \leq 0\). Plotting values like \( x = -1 \) gives you \( g(-1) = \sqrt{1} = 1 \) and for \( x = -4 \), \( g(-4) = \sqrt{4} = 2 \). The graph starts at the point \( (0,0) \) and creates a smooth curve leftwards, showcasing a half parabola. This visual representation aids in understanding the function's behavior across different domains.
Reflection Across Axes
Reflection across axes in graphing refers to flipping a graph over a particular axis. For the function \( g(x) = \sqrt{-x} \), you can interpret it as a reflection of the graph \( f(x) = \sqrt{x} \) across the y-axis. The "classic" square root function, \( f(x) = \sqrt{x} \), is known for its rightward facing arc that starts at \( (0,0) \). Reflecting it over the y-axis means that each point \((x, y)\) in \(\sqrt{x}\) is transformed to \((-x, y)\) in \(\sqrt{-x}\). This transformation creates a leftward-facing curve, highlighting the symmetrical properties of the function.

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Most popular questions from this chapter

In Exercises \(5-30,\) find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. $$ y=x+\frac{1}{10} \sin 30 x $$

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side \(x\) at each corner and then folding up the sides as in the figure. Express the volume \(V\) of the box as a function of \(x .\)

A balloon's volume \(V\) is given by \(V=s^{2}+2 s+3 \mathrm{cm}^{3},\) where \(s\) is the ambient temperature in 'C. The ambient temperature \(s\) at time \(t\) minutes is given by \(s=2 t-3^{\circ} \mathrm{C} .\) Write the balloon's volume \(V\) as a function of time \(t\)

Exercises \(59-68\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=1-x^{3}, \quad \text { stretched horizontally by a factor of } 2 $$

Let \(f(x)=x-3, \quad g(x)=\sqrt{x}, \quad h(x)=x^{3},\) and \(j(x)=2 x .\) Express each of the functions in Exercises 11 and 12 as a composition involving one or more of \(f, g, h,\) and \(j\) $$\begin{array}{ll}{\text { a. } y=\sqrt{x}-3} & {\text { b. } y=2 \sqrt{x}} \\\ {\text { c. } y=x^{1 / 4}} & {\text { d. } y=4 x} \\ {\text { e. } y=\sqrt{(x-3)^{3}}} & {\text { f. } y=(2 x-6)^{3}}\end{array}$$
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