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Chapter 2: Problem 28

Find the rules for the composite functions \(f \circ g\) and \(g \circ f\). \(f(x)=2 \sqrt{x}+3 ; g(x)=x^{2}+1\)

Short Answer

Expert verified
The rules for the composite functions are: \(f(g(x)) = 2\sqrt{x^2 + 1} + 3\) and \(g(f(x)) = 4x + 12\sqrt{x} + 10\).

Step by step solution

01

Find the composite function \(f \circ g\)

To find the composite function \(f \circ g\), we need to substitute \(g(x)\) into \(f(x)\). This means we replace the \(x\) in the function \(f(x)\) with the entire function \(g(x)\): \[f(g(x)) = 2\sqrt{g(x)} + 3\] Now, substitute the function \(g(x) = x^2 + 1\) into the equation: \[f(g(x)) = 2\sqrt{x^2 + 1} + 3\] The rule for the composite function \(f \circ g\) is \(f(g(x)) = 2\sqrt{x^2 + 1} + 3\).
02

Find the composite function \(g \circ f\)

To find the composite function \(g \circ f\), we need to substitute \(f(x)\) into \(g(x)\). This means we replace the \(x\) in the function \(g(x)\) with the entire function \(f(x)\): \[g(f(x)) = (f(x))^2 + 1\] Now, substitute the function \(f(x) = 2\sqrt{x} + 3\) into the equation: \[g(f(x)) = (2\sqrt{x} + 3)^2 + 1\] To simplify the expression, we will expand the square by using the formula \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = 2\sqrt{x}\) and \(b = 3\): \[g(f(x)) = (2\sqrt{x})^2 + 2(2\sqrt{x})(3) + 3^2 + 1\] \[g(f(x)) = 4x + 12\sqrt{x} + 9 + 1\] \[g(f(x)) = 4x + 12\sqrt{x} + 10\] The rule for the composite function \(g \circ f\) is \(g(f(x)) = 4x + 12\sqrt{x} + 10\).

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Key Concepts

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

Function Composition
Function composition is like stacking two functions on top of each other. You take the output of one function and use it as the input for another. Here, we find two composite functions: \(f \circ g\) and \(g \circ f\).
  • For \(f \circ g\), we plug the function \(g(x)\) into \(f(x)\). This means wherever there's an \(x\) in \(f(x)\), we replace it with \(g(x)\), giving us \(f(g(x)) = 2\sqrt{x^2 + 1} + 3\).
  • For \(g \circ f\), we insert \(f(x)\) into \(g(x)\), replacing the \(x\) in \(g(x)\) with \(f(x)\). This results in \(g(f(x)) = 4x + 12\sqrt{x} + 10\).
Function composition helps us create new functions by combining existing ones, offering powerful ways to model complex relationships.
Mathematical Functions
Mathematical functions are expressions that relate an input to an output. In this exercise, we work with two basic functions, \(f(x)\) and \(g(x)\).
  • The function \(f(x) = 2 \sqrt{x} + 3\) contains a square root followed by a linear term, scaling the input \(x\) and adding 3.
  • The function \(g(x) = x^2 + 1\) is a quadratic expression, squaring the input and adding 1, creating a parabolic shape.
Functions are like machines that take an input, process it according to a rule, and give an output, helping us analyze and interpret various real-world situations.
Algebraic Expressions
Algebraic expressions are combinations of symbols and numbers that define a mathematical function. Simplifying these expressions is crucial for understanding. Let's break it down from the exercise:
  • In finding \(g \circ f\), after inserting \(f(x)\) into \(g(x)\), we used expansion to simplify \((2\sqrt{x} + 3)^2 + 1\).
  • Applying the formula \((a + b)^2 = a^2 + 2ab + b^2\) helped us transform the expression into \(4x + 12\sqrt{x} + 10\).
These transformations reveal how algebraic identities and operations streamline our understanding of function compositions, leading to clear and concise expressions.

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Most popular questions from this chapter

Gift cards have increased in popularity in recent years. Consumers appreciate gift cards because they get to select the present they like. The U.S. sales of giff cards (in billions of dollars) is approximated by \(S(t)=-0.6204 t^{3}+4.671 t^{2}+3.354 t+47.4 \quad(0 \leq t \leq 5)\) in year \(t\), where \(t=0\) corresponds to 2003 . a. What were the sales of gift cards for 2003 ? b. What were the sales of gift cards in 2008 ?

BROADBAND VERSUS DIAL-UP The number of U.S. broadband Internet households (in millions) between the beginning of \(2004(t=0)\) and the beginning of \(2008(t=4)\) was estimated to be $$ f(t)=6.5 t+33 \quad(0 \leq t \leq 4) $$ Over the same period, the number of U.S. dial-up Internet households (in millions) was estimated to be $$ g(t)=-3.9 t+42.5 \quad(0 \leq t \leq 4) $$ a. Sketch the graphs of \(f\) and \(g\) on the same set of axes. b. Solve the equation \(f(t)=g(t)\) and interpret your result.

For each pair of supply and demand equations where \(x\) represents the quantity demanded in units of a thousand and \(p\) the unit price in dollars, find the equilibrium quantity and the equilibrium price. \(p=60-2 x^{2}\) and \(p=x^{2}+9 x+30\)

A workcenter system purchased at a cost of \(\$ 60,000\) in 2007 has a scrap value of \(\$ 12,000\) at the end of 4 yr. If the straight-line method of depreciation is used, a. Find the rate of depreciation. b. Find the linear equation expressing the system's book value at the end of \(t\) yr. c. Sketch the graph of the function of part (b). d. Find the system's book value at the end of the third year.

The sales of DVD players in year \(t\) (in millions of units) is given by the function $$ f(t)=5.6(1+t) \quad(0 \leq t \leq 3) $$ where \(t=0\) corresponds to 2001 . Over the same period, the sales of VCRs (in millions of units) is given by $$ g(t)=\left\\{\begin{array}{ll} -9.6 t+22.5 & \text { if } 0 \leq t \leq 1 \\ -0.5 t+13.4 & \text { if } 1
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