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Chapter 1: Problem 42

Find formulas for \(f \circ g\) and \(g \circ f,\) and state the domains of the functions. $$f(x)=\sqrt{x-3}, g(x)=\sqrt{x^{2}+3}$$

Short Answer

Expert verified
\(f \circ g(x) = \sqrt{\sqrt{x^2 + 3} - 3}\), domain: \(x \leq -\sqrt{6}\) or \(x \geq \sqrt{6}\). \(g \circ f(x) = \sqrt{x}\), domain: \(x \geq 3\).

Step by step solution

01

Understand the composition of functions

The composition of functions is an operation where one function is applied to the result of another function. For two functions, say \(f(x)\) and \(g(x)\), the composition \(f \circ g\) is defined as \(f(g(x))\) and \(g \circ f\) as \(g(f(x))\). We need to find these compositions and their domains.
02

Find \(f \circ g(x)\)

To find \(f \circ g(x)\), substitute \(g(x) = \sqrt{x^2 + 3}\) into \(f(x)\). Do this by replacing \(x\) in \(f(x) = \sqrt{x - 3}\) with \(\sqrt{x^2 + 3}\) leading to:\[ f(g(x)) = f(\sqrt{x^2 + 3}) = \sqrt{\sqrt{x^2 + 3} - 3} \]
03

Domain of \(f \circ g(x)\)

The domain of \(f \circ g(x)\) requires \(\sqrt{x^2 + 3} - 3 \geq 0\), which solves to \(x^2 + 3 \geq 9\). Solve for \(x\):\[ x^2 \geq 6 \]So, the domain is \( x \leq -\sqrt{6} \) or \( x \geq \sqrt{6} \).
04

Find \(g \circ f(x)\)

To find \(g \circ f(x)\), substitute \(f(x) = \sqrt{x - 3}\) into \(g(x)\). Do this by replacing \(x\) in \(g(x) = \sqrt{x^2 + 3}\) with \(\sqrt{x - 3}\), leading to:\[ g(f(x)) = g(\sqrt{x - 3}) = \sqrt{(\sqrt{x - 3})^2 + 3} = \sqrt{x - 3 + 3} = \sqrt{x} \]
05

Domain of \(g \circ f(x)\)

The domain of \(g \circ f(x)\) requires both \(f(x) = \sqrt{x - 3}\) to be defined and \(g(\sqrt{x - 3}) = \sqrt{x}\) to be defined. Therefore, \(x - 3 \geq 0\) which implies \(x \geq 3\), and from \(\sqrt{x}\), \(x \geq 0\). Thus, the domain is \(x \geq 3\).

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Key Concepts

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

Understanding Functions
Functions are like machines that take an input, process it according to a specific rule, and produce an output. Each function is denoted as a mathematical notation, typically written as \( f(x) \), where \( f \) represents the function and \( x \) is the input variable.
For example, \( f(x) = \sqrt{x-3} \) is a function where you subtract 3 from \( x \) and take the square root of the result. If you provide an input value for \( x \), you can determine its output by following the function's rule.
  • The input to a function is often called the 'independent variable.'
  • The output from a function is called the 'dependent variable.'
Functions are fundamental in math because they describe the relationship between variables. They are used across various fields like physics, engineering, and economics to model real-life phenomena.
Exploring the Domain of a Function
The domain of a function refers to all the possible input values that the function can accept. It's crucial to determine the domain to know where the function exists in mathematical terms.
Each function has its specific domain. For example, with \( f(x) = \sqrt{x-3} \), the expression under the square root, \( x-3 \), must be greater than or equal to zero so that we have a real number result from the square root. Thus, the domain is \( x \geq 3 \).
  • Domains are determined by considering mathematical operations: for instance, no division by zero is allowed.
  • The nature of square roots requires that expressions under the root be non-negative for real numbers.
Knowing the domain helps in graphing the function and figuring out its possible applications.
The Nature of Square Root Functions
A square root function is one where the principal operation involves a square root. It follows the form \( f(x) = \sqrt{x} \), meaning it will output the non-negative square root of its input.
Consider a function like \( g(x) = \sqrt{x^2 + 3} \). Here, regardless of the value of \( x \), the expression inside the square root is always non-negative, hence \( g(x) \) is defined for all real numbers.
  • Square root functions are only defined for non-negative expressions under the root.
  • They are written in simplified form as square roots yield real numbers only when the input is non-negative.
Understanding this is key to interpreting mathematical functions correctly, especially when composing multiple functions together.

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Most popular questions from this chapter

Find a formula for \(f \circ g \circ h\) $$f(x)=x^{2}+1, g(x)=\frac{1}{x}, h(x)=x^{3}$$

Sketch the graph of the equation by translating. reflecting, eompressing. and stretching the graph of \(y=\sqrt[3]{x}\) appropriately. and then use a graphing utility to confirm that your sketch is correct. $$y=\sqrt[3]{x-2}-3$$

(a) Use a graphing utility to generate the trajectory of a particle whose equations of motion over the time interval $$\begin{aligned}0 \leq & t \leq 5 \text { are } \\\x &=6 t-\frac{1}{2} t^{3}, \quad y=1+\frac{1}{2} t^{2}\end{aligned}$$ (b) Make a table of \(x\) - and \(y\) -coordinates of the particle at times \(t=0,1,2,3,4,5\) (c) Al what times is the particle on the \(y\) -axis? (d) During what time interval is \(y<5 ?\) (e) At what time is the \(x\) -coordinate of the particle maximum?

(a) Find parametric equations for the ellipse that is centered at the origin and has intercepts (4,0),(-4,0),(0,3) and (0,-3). (b) Find parametric equations for the ellipse that results by translating the ellipse in part (a) so that its center is at (-1,2). (c) Confirm your results in parts (a) and (b) using a graphing utility.

What do you think the graph of \(y=\sin (1 / x)\) looks like? Test your conclusion using a graphing utility. [Sieggestion: Examine the graph on a succession of smaller and smaller intervals centered at \(x=0.1\)
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