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Chapter 3: Problem 31

State the instructions of the function in words as in Example 9 $$g(y)=6 y-y^{3}$$

Short Answer

Expert verified
For \( g(y) \), multiply \( y \) by 6 and subtract \( y^3 \).

Step by step solution

01

Understand the Function

The function is given as \( g(y) = 6y - y^3 \). This means that for any input \( y \), the output is calculated by multiplying \( y \) by 6 and then subtracting the cube of \( y \).
02

Follow the Instructions in Words

According to the problem, the instructions of the function need to be stated in words, similar to Example 9. For any given \( y \), first multiply \( y \) by 6, which gives \( 6y \). Then, calculate the cube of \( y \) to get \( y^3 \). Finally, subtract \( y^3 \) from \( 6y \).
03

Write Down the Function's Instruction

In words, the function \( g(y) = 6y - y^3 \) can be expressed as: To evaluate \( g(y) \), triple the input \( y \) twice (since 6 is 3 + 3), and subtract the cube of \( y \) from it.

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Key Concepts

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

Cubics
Cubics are polynomial functions where the highest degree of variable is three. Simply speaking, a cubic function will have a term like \( y^3 \). This term means that whatever number you put in for \( y \), you multiply it by itself twice more: \( y \times y \times y \). Cubic terms create a unique curve when graphed, often known as a cubic curve, having three possible roots.
  • They might rise to infinity or fall to negative infinity depending on the leading coefficient.
  • If the leading term is negative, it will slope downwards.
  • If the leading term is positive, it will slope upwards.
Thinking about \( y^3 \) as a separate piece of the overall function helps to see how each part contributes to the final outcome when combined with other terms.
Multiplication in Functions
Multiplication plays a central role in function operations. In polynomial functions like the one we discussed, multiplication helps in scaling numbers. When the function \( g(y) = 6y - y^3 \) uses multiplication, it modifies the value of input \( y \) by a constant factor. Here鈥檚 how it works step by step:
  • Take the input variable \( y \).
  • Multiply \( y \) by 6 to scale it.
  • This linear operation is combined with a non-linear operation like cubing, affecting the function's overall behavior.
This combination of linear and non-linear operations is key in polynomial functions, leading to interesting graphs and variable behaviors.
Function Notation
Function notation is a concise way of showing the input-output relationship in mathematical functions. It describes how every input in the domain produces an output in the range. Using function notation, such as \( g(y) = 6y - y^3 \), helps to quickly express a rule without writing verbose explanations. Here are some essential things about function notation:
  • The function name (like \( g \)) depicts the purpose, which could be anything, here it's just a labeling.
  • The variable in the parentheses \((y)\) represents the input.
  • The equation on the right (\( 6y - y^3 \)) defines the operation that transforms \( y \) into an output.
Recognizing function notation is crucial. It simplifies expressing and manipulating complex operations logically and clearly, helping students solve problems step by step.

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Most popular questions from this chapter

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