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Chapter 9: Problem 4

After reading this section, write out the answers to these questions. Use complete sentences. How is the order of operations related to composition of functions?

Short Answer

Expert verified
Both the order of operations and composition of functions require performing operations in a specific sequence to get correct results.

Step by step solution

01

- Understand Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed to accurately solve an expression.
02

- Understand Composition of Functions

Composition of functions involves applying one function to the result of another function. If you have two functions, say f(x) and g(x), the composition is written as \(f(g(x))\), meaning you first apply g to x, and then f to the result of g(x).
03

- Relate Order of Operations to Composition of Functions

The order of operations is related to the composition of functions because both involve a specific sequence. In the composition of functions, you must evaluate the inner function first (just like solving operations inside parentheses first in PEMDAS), then apply the outer function to the result.
04

- Formulate the Answer

The order of operations is related to the composition of functions in that both require a specific sequence to be followed to get the correct result. For composition, you perform the inner function first, then the outer function, similar to solving operations inside parentheses before moving on to other operations in PEMDAS.

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Key Concepts

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

Order of Operations
To solve mathematical expressions correctly, you must follow the order of operations. This ensures you perform operations in the right sequence, avoiding mistakes. The order can be remembered using the acronym PEMDAS:
  • Parentheses: Solve expressions inside parentheses first.
  • Exponents: Next, calculate exponents (powers and roots).
  • Multiplication and Division: Then, perform multiplication and division from left to right.
  • Addition and Subtraction: Lastly, do addition and subtraction from left to right.
The order is crucial to getting the correct answer because it dictates which parts of the expression to simplify first. For instance, in the expression 3 + 2 脳 (8 - 5)2, you would:
  • First handle the parentheses: 3 + 2 脳 32.
  • Next, calculate the exponent: 3 + 2 脳 9.
  • Then, do the multiplication: 3 + 18.
  • Finally, perform the addition: 21.
Not following PEMDAS can lead to incorrect results, which makes understanding this sequence fundamental in mathematics.
PEMDAS
PEMDAS stands for: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This acronym helps in remembering the exact order to perform operations in a mathematical expression. Each component signifies a different step in solving the expression:
  • Parentheses: Always start with expressions within parentheses.
  • Exponents: Follow up by calculating any exponents.
  • Multiplication and Division: Conduct these operations next, moving from left to right.
  • Addition and Subtraction: Lastly, complete the addition and subtraction, from left to right.
For example, in (2 + 3) 脳 42 - 6/3:
  • Solve inside parentheses first: 5 脳 42 - 6/3.
  • Calculate exponents: 5 脳 16 - 6/3.
  • Do multiplication and division: 80 - 2.
  • Perform addition and subtraction: 78.
Using PEMDAS guarantees order and accuracy in solving expressions, helping you avoid mistakes and ensuring your answer is correct.
Function Evaluation
Function evaluation refers to finding the output of a function for a specific input. If you have a function, say f(x), and want to determine its value at a specific point, you substitute the given input into the function.
For example:
If f(x) = x2 + 3x,
to find f(2), you:
  • Substitute 2 for x: f(2) = 22 + 3 脳 2.
  • First calculate the exponent: f(2) = 4 + 3 脳 2.
  • Then perform the multiplication: f(2) = 4 + 6.
  • Finally, do the addition: f(2) = 10.
In another case, if g(y) = 2y - 5 and you want to find g(3):
  • Substitute 3 for y: g(3) = 2 脳 3 - 5.
  • Then do the multiplication: g(3) = 6 - 5.
  • Lastly, perform the subtraction: g(3) = 1.
Accurate function evaluation is essential as it helps in understanding how functions behave for different inputs. It also provides a solid foundation for more complex operations, like composition of functions.

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Most popular questions from this chapter

Solve each problem. See Examples 5鈥�7. Bicycle gear ratio. A bicycle's gear ratio \(G\) varies jointly with the number of teeth on the chain ring \(N\) (by the pedals ) and the diameter of the wheel \(d,\) and inversely with the number of teeth on the \(\operatorname{cog} c\) (on the rear wheel). A bicycle with 27 -inch-diameter wheels, 26 teeth on the cog, and 52 teeth on the chain ring has a gear ratio of 54 a) Find a formula that expresses the gear ratio as a function of \(N, d,\) and \(c .\) b) What is the gear ratio for a bicycle with 26 -inchdiameter wheels, 42 teeth on the chain ring, and 13 teeth on the cog? c) A five-speed bicycle with 27 -inch-diameter wheels and 44 teeth on the chain ring has gear ratios of \(52,59\) \(70,79,\) and \(91 .\) Find the number of teeth on the cog (a whole number) for each gear ratio. d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases?

Classify each function as either a linear, constant, quadratic, square-root, or absolute value function. $$f(x)=4$$

Let \(f(x)=x^{2}\) and \(g(x)=x+5 .\) Determine whether each of these statements is true or false. $$(f-g)(0)=5$$

Find \(f^{-1}\). Check that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\) Strategy for Finding \(f^{-1}\) by Switch-and Solve. $$f(x)=\frac{2}{x+1}$$

Classify each function as either a linear, constant, quadratic, square-root, or absolute value function. $$f(x)=\sqrt{x-3}$$
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