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Chapter 3: Problem 30

Let \(a(x)=x^{2}, b(x)=|x|,\) and \(c(x)=3 x-1 .\) Express each of the following functions as a composition of two of the given functions. (a) \(f(x)=(3 x-1)^{2}\) (c) \(h(x)=3 x^{2}-1\) (b) \(g(x)=|3 x-1|\)

Short Answer

Expert verified
(a) \(f(x) = a(c(x))\); (c) \(h(x) = c(a(x))\); (b) \(g(x) = b(c(x))\).

Step by step solution

01

Analyze function \( f(x) = (3x-1)^2 \)

To express \( f(x) = (3x-1)^2 \) as a composition of two functions from \( a(x), b(x), \) and \( c(x) \), first identify that \((3x-1)\) resembles \(c(x) = 3x - 1\). The squared term suggests \(a(x) = x^2\) might be involved. Hence, the composition is likely \(f(x) = a(c(x))\).
02

Express \( h(x) = 3x^2 - 1 \)

Here, \( h(x) = 3x^2 - 1\) can be analyzed as first applying \( a(x) = x^2 \), followed by \( c(x) = 3x - 1 \) on the result. Therefore, \( h(x) = c(a(x)) \).
03

Describe \( g(x) = |3x - 1| \)

For \( g(x) = |3x - 1| \), notice that \( |3x - 1| \) involves taking the absolute value of \( c(x) = 3x - 1 \). This function can be expressed as \( g(x) = b(c(x)) \).

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Key Concepts

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

Quadratic Functions
Quadratic functions are an important type of polynomial function characterized by their specific form: \( y = ax^2 + bx + c \). They graph as a parabola, which can open upwards or downwards, depending on the sign of the coefficient 'a'. Here are the key features:
  • The vertex is the peak or the lowest point of the parabola, depending on the direction it opens.
  • The axis of symmetry is a vertical line that runs through the vertex, splitting the graph into two mirror-image parts.
  • The roots or zeros of the quadratic function are the x-values where the graph crosses the x-axis. These are the solutions to the quadratic equation \( ax^2 + bx + c = 0 \).
In function composition, as in the exercise, we use quadratic functions to build complex expressions, like \( f(x) = (3x-1)^2 \). Here, \((3x-1)\) reshapes and repositions the parabola of \( x^2 \). Understanding the roles of each component helps reveal how transformations affect the quadratic function.
Absolute Value
The absolute value function, represented as \( b(x) = |x| \), is a special type of function that outputs the non-negative value of any input 'x'. Its graph is a "V" shape, centered at the origin with the following properties:
  • The function is always non-negative.
  • It is piecewise linear, meaning its graph is comprised of two linear segments.
  • The absolute value indicates the distance of a number from zero on a number line, regardless of direction.
In the exercise problem \( g(x) = |3x-1| \), the function \( |x| \) is used in composition to handle the expression \( c(x) = 3x-1 \). This means all negative results produced by \( 3x-1 \) are flipped to positive. This transformation changes the output but preserves numerical distance, reshaping inputs into non-negative outputs.
Linear Functions
Linear functions, such as \( c(x) = 3x - 1 \), are straight-line functions characterized by their slope and y-intercept. They are fundamental to understanding more complex functions because they represent constant rates of change. Here's what you need to know:
  • The standard form is \( y = mx + b \), where 'm' is the slope indicating the steepness, and 'b' is the y-intercept where the line crosses the y-axis.
  • The slope 'm' shows how much 'y' changes for each unit increase in 'x'.
  • Graphically, a linear function is a straight line with no curves or bends.
In function composition, linear functions serve as a building block to alter or transform mathematical expressions. For instance, the function \( f(x) = (3x-1)^2 \) starts by applying the linear function \( c(x) = 3x - 1 \). This operation shifts and scales the input before applying further operations, like squaring, to achieve the desired result.

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Most popular questions from this chapter

Let \(f(x)=x^{2}-6 x\). In each case, find all real numbers \(x\) (if any) that satisfy the given equation. (a) \(f(x)=16\) (b) \(f(x)=-10\) (c) \(f(x)=-9\)

Suppose that an oil spill in a lake covers a circular area and that the radius of the circle is increasing according to the formula \(r=f(t)=15+t^{1.65},\) where \(t\) represents the number of hours since the spill was first observed and the radius \(r\) is measured in meters. (Thus when the spill was first discovered, \(t=0 \mathrm{hr}\), and the initial radius was \(\left.r=f(0)=15+0^{1.65}=15 \mathrm{~m} .\right)\) (a) Let \(A(r)=\pi r^{2},\) as in Example \(5 .\) Compute a table of values for the composite function \(A \circ f\) with \(t\) running from 0 to 5 in increments of \(0.5 .\) (Round each output to the nearest integer.) Then use the table to answer the questions that follow in parts (b) through (d). (b) After one hour, what is the area of the spill (rounded to the nearest \(10 \mathrm{~m}^{2}\) )? (c) Initially, what was the area of the spill (when \(t=0\) )? Approximately how many hours does it take for this area to double? (d) Compute the average rate of change of the area of the spill from \(t=0\) to \(t=2.5\) and from \(t=2.5\) to \(t=5\). Over which of the two intervals is the area increasing faster?

Indicate how iteration is used in finding roots of numbers and roots of equations. (The functions that are given in each exercise were determined using Newton's method, a process studied in calculus.) Let \(f(x)=\frac{2 x^{3}+7}{3 x^{2}}\). (a) Compute the first ten iterates of \(x_{0}=1\) under the function \(f .\) What do you observe? (b) Evaluate the expression \(\sqrt[3]{7}\) and compare the answer to your results in part (a). What do you observe? (c) It can be shown that for any positive number \(x_{0}\), the iterates of \(x_{0}\) under the function \(f(x)=\frac{2 x^{3}+7}{3 x^{2}}\) always approach the number \(\sqrt[3]{7} .\) Looking at your results in parts (a) and (b), which is the first iterate that agrees with \(\sqrt[3]{7}\) through the first three decimal places? Through the first eight decimal places?

Suppose that in a certain biology lab experiment, the number of bacteria is related to the temperature \(T\) of the environment by the function $$N(T)=-2 T^{2}+240 T-5400 \quad(40 \leq T \leq 90)$$ Here, \(N(T)\) represents the number of bacteria present when the temperature is \(T\) degrees Fahrenheit. Also, suppose that \(t\) he experiment begins, the temperature is given by $$T(t)=10 t+40 \quad(0 \leq t \leq 5)$$ (a) Compute \(N[T(t)]\) (b) How many bacteria are present when \(t=0\) hr? When \(t=2 \mathrm{hr} ?\) When \(t=5 \mathrm{hr} ?\)

Let \(P\) be a point with coordinates \((a, b),\) and assume that \(c\) and \(d\) are positive numbers. (The condition that \(c\) and \(d\) are positive isn't really necessary in this problem, but it will help you to visualize things.) (a) Translate the point \(P\) by \(c\) units in the \(x\) -direction to obtain a point \(Q,\) then translate \(Q\) by \(d\) units in the y-direction to obtain a point \(R\). What are the coordinates of the point \(R ?\) (b) Translate the point \(P\) by \(d\) units in the \(y\) -direction to obtain a point \(S,\) then translate \(S\) by \(c\) units in the \(x\) -direction to obtain a point \(T .\) What are the coordinates of the point \(T ?\) (c) Compare your answers for parts (a) and (b). What have you demonstrated? (Answer in complete sentences.)
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