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Chapter 1: Problem 38

Give the domain of the function and sketch the graph. $$f(x)=\sqrt{9-x^{3}}$$

Short Answer

Expert verified
The domain of the function \(f(x)=\sqrt{9-x^{3}}\) is \(-\infty \leq x \leq \sqrt[3]{9}\). The graph of the function starts from the point \((\sqrt[3]{9}, 0)\) and falls as x decreases, never crossing the x-axis.

Step by step solution

01

Determine the Domain of Function

The domain of the function is determined by setting the argument of the square root, \(9-x^{3}\), to be greater than or equal to 0. So, solve \(9-x^{3} \geq 0\). This implies that \(x^{3} \leq 9\). Taking cube root both side to isolate x, we find that \(x \leq \sqrt[3]{9}\). Hence, the domain of the function is all real numbers less than or equal to \(\sqrt[3]{9}\). i.e., \(-\infty \leq x \leq \sqrt[3]{9}\).
02

Graph the Function

To graph the function, one would first need to sketch the function \(x^{3}\) and then apply the transformation inherent in the `square root`. It can be noted that for \(x^{3}\), as x increases, the value of \(x^{3}\) also increases. Now, the argument of the square root function should be \(9-x^{3} \geq 0\). Where \(9-x^{3}=0\), i.e., at \(x = \sqrt[3]{9}\), will be the point where graph of this function starts. This ensures the square root is never undefined. Thus, the graph of the function will start from the point \((\sqrt[3]{9}, 0)\) and then fall as x decreases. It will never cross the x-axis because the square root function always returns positive values or zero.

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Key Concepts

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

Graphing Functions
Graphing functions helps us visualize how they behave. We can see where they are increasing or decreasing, and identify special points such as intercepts. To do this, it's useful to understand transformations, as they determine how the function's graph changes with respect to its parent function.
  • Start with a basic graph: For example, the graph of the cubic function, \(x^3\)
  • Apply transformations: In \(9-x^3\), we subtract the cubic to create a new curve.
  • Consider domain constraints: As given, we only graph where \(9-x^3 \geq 0\). This gives visibility to only the relevant part of the graph.

By sketching the graph step-by-step, you can see how each piece of the equation affects the final shape.
Square Root Function
The square root function, represented as \(\sqrt{x}\), is a basic yet important function in mathematics. It only produces real numbers when its argument is non-negative.
  • Domain: For \(\sqrt{9-x^3}\), the domain is restricted to ensure the expression under the root is non-negative.
  • Graph Shape: The graph of a square root function starts at the point where the argument is zero and increases slowly, creating a gentle curve.

This is why you see the square root function begin at zero and gradually rise, never dipping below the x-axis.
Cubic Root
The cubic root function is denoted by \(\sqrt[3]{x}\). Unlike square roots, cubic roots can take negative and positive values, allowing the domain to be all real numbers.
  • Solve Cubic Equations: To find domains or intercepts, use the cubic root to simplify and explore parts of a function.
  • Equal Passages: Since it crosses zero, it's helpful in breaking down inequalities such as \(x^3 \leq 9\).

Understanding what a cubic root implies in each situation clarifies how transformations affect graph shapes and domain restrictions.
Inequalities in Functions
Inequalities play a crucial role in determining valid inputs for functions, especially when they involve roots.
  • Domains: When setting inequalities for \(9-x^3 \geq 0\), we determine valid domains for the output.
  • Solve Step by Step: Break down complex inequalities into simpler equations such as \(x^3 \leq 9\), then apply functions like cubic roots.

Through inequality solving, we set effective constraints, helping us sketch accurate graphs and set precise ranges for function definitions.

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Most popular questions from this chapter

Find \(f \circ g\) and \(g \circ f\). $$f(x)=x^{3}+1, g(x)=\sqrt[3]{x-1}$$

Find an equation for the line that passes through the point (2,-3) and is perpendicular to the line \(2 x-3 y=6\)

Suppose that \(f\) and \(g\) arc even functions. What can you conclude about \(f \cdot g ?\)

Form the compositions \(f \circ g\) and \(g \circ f,\) and specify the domain of each of these combinations. $$f(x)=\sqrt{x+1}, \quad g(x)=x^{2}-5$$

Find \(g\) such that \(f \circ g=F\) given that $$f(x)=x^{2}+1 , F(x)=\left(2 x^{3}-1\right)^{2}+1$$
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