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Chapter 2: Problem 10

Find (a) \((f \circ g)(x)\) (b) \((g \circ f)(x)\) (c) \((f \circ f)(x)\) (d) \((g \circ g)(x)\) $$f(x)=3 x^{2}, \quad g(x)=x-1$$

Short Answer

Expert verified
(a) \(3x^2 - 6x + 3\), (b) \(3x^2 - 1\), (c) \(27x^4\), (d) \(x - 2\).

Step by step solution

01

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

To find \((f \circ g)(x)\), we need to substitute \(g(x) = x-1\) into \(f(x)\). Start by replacing \(x\) in \(f(x) = 3x^2\) with \(g(x)\).\[(f \circ g)(x) = f(g(x)) = f(x-1)\]Now calculate:\[f(x-1) = 3(x-1)^2 = 3(x^2 - 2x + 1) = 3x^2 - 6x + 3\]Thus, \((f \circ g)(x) = 3x^2 - 6x + 3\).
02

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

To find \((g \circ f)(x)\), we substitute \(f(x) = 3x^2\) into \(g(x)\). Replace \(x\) in \(g(x) = x-1\) with \(f(x)\).\[(g \circ f)(x) = g(f(x)) = g(3x^2)\]Now calculate:\[g(3x^2) = (3x^2) - 1 = 3x^2 - 1\]Thus, \((g \circ f)(x) = 3x^2 - 1\).
03

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

To find \((f \circ f)(x)\), substitute \(f(x) = 3x^2\) into itself.\[(f \circ f)(x) = f(f(x)) = f(3x^2)\]Now calculate:\[f(3x^2) = 3(3x^2)^2 = 3(9x^4) = 27x^4\]Thus, \((f \circ f)(x) = 27x^4\).
04

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

To find \((g \circ g)(x)\), substitute \(g(x) = x-1\) into itself.\[(g \circ g)(x) = g(g(x)) = g(x-1)\]Now calculate:\[g(x-1) = (x-1) - 1 = x - 2\]Thus, \((g \circ g)(x) = x - 2\).

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Key Concepts

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

Understanding Composite Functions
Composite functions involve combining two or more functions to create a new function. This process is much like putting ingredients together to make a cake. You have two functions, say \(f(x)\) and \(g(x)\), and instead of working with them separately, you use the output of one as the input for the other.
  • Composition Notation: The composition of \(f(x)\) and \(g(x)\) is written as \((f \circ g)(x)\). This means you apply \(g(x)\) first, then \(f(x)\) to the result.
  • Important Order: Remember, the order matters! \((f \circ g)(x)\) is not the same as \((g \circ f)(x)\). Think of it as following a sequence of steps where the order can affect the result.
In our exercise, we calculated composite functions like \((f \circ g)(x) = 3x^2 - 6x + 3\) by plugging \(x-1\) from \(g(x)\) into \(f(x)\). We did similar substitutions to find \((g \circ f)(x)\), \((f \circ f)(x)\), and \((g \circ g)(x)\), demonstrating the versatile nature of these functions.
Exploring Quadratic Functions
Quadratic functions are a type of polynomial function that can be written in the format \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The key feature of a quadratic function is the \(x^2\) term, which creates a parabola when graphed.
  • Standard Form: The most common quadratic function form is \(f(x) = 3x^2\) in our exercise, indicating a parabola opening upwards.
  • Vertex: The vertex of this parabola is at the origin \((0,0)\) because there are no linear or constant terms altering its position.
The quadratic function \(f(x) = 3x^2\) in our example is straightforward, with all results remaining simple due to the lack of \(b\) and \(c\) terms. When substituted in compositions like \((f \circ f)(x)\), it demonstrates how quadratic functions can become more complex through re-composition.
Performing Algebraic Operations
Algebraic operations are the basic mathematical procedures of addition, subtraction, multiplication, and division involving numbers, variables, and their combinations. Understanding these operations is essential for solving and manipulating equations.
  • Substitution: This involves replacing a variable with a given expression. For composite functions, we substitute one function into another.
  • Simplification: This is the process of rewriting expressions as simply as possible by performing arithmetic operations and combining like terms.
During the composite function calculations, we used substitution and simplification to transform functions results, such as turning \(3(x-1)^2\) into \(3x^2 - 6x + 3\). Recognizing these basic operations allows for creating more complex functions while keeping expressions manageable and understandable.

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Most popular questions from this chapter

Flights of leaping animals typically have parabolic paths. The figure on the next page illustrates a frog jump superimposed on a coordinate plane. The length of the leap is 9 feet, and the maximum height off the ground is 3 feet. Find a standard equation for the path of the frog.

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Cable TY fee A cable television firm presently serves 8000 households and charges \(\$ 50\) per month. A marketing survey indicates that each decrease of \(\$ 5\) in the monthly charge will result in 1000 new customers. Let \(R(x)\) denote the total monthly revenue when the monthly charge is \(x\) dollars. (a) Determine the revenue function \(R\) (b) Sketch the graph of \(R\) and find the value of \(x\) that results in maximum monthly revenue.
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