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Chapter 2: Problem 66

Let \(f(x)=3 x^{2}-x\) and \(g(x)=4 x-2 .\) Find the following. $$f(w)$$

Short Answer

Expert verified
f(w) = 3w^2 - w

Step by step solution

01

- Identify the given function

The function provided is \(f(x) = 3x^2 - x\). This function will be used to find the value of \(f(w)\).
02

- Substitute the value

Replace \(x\) with \(w\) in the function \(f(x)\). This means we will substitute \(w\) into every instance of \(x\) in the function.
03

- Substitute and simplify

Substitute \(w\) for \(x\) in \(f(x) = 3x^2 - x\). This gives us: \(f(w) = 3w^2 - w\). No simplification is needed beyond this as it is already in its simplest form.

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Key Concepts

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

Function Notation
Function notation is a way to write functions in a more readable form. Instead of writing equations with variables like we do in algebra, functions are written with notations like: \(f(x) = 3x^2 - x\).

In this notation:
  • The letter \(f\) represents the function name.
  • The variable \(x\) inside the parentheses is the input to the function.
Interestingly, you can use any letter or even words as long as they follow the format \(function\text{(input)} = expression\). This makes it simple to see what a function does to its input by just looking at it. When we change the input, we can see how it affects the output directly.
Substitution in Functions
Substitution in functions is similar to replacing a variable in an equation with a specific value. This concept is crucial when dealing with function notation. To illustrate, consider the function \(f(x) = 3x^2 - x\).

When we need to find the value of \(f(w)\), we substitute all instances of \(x\) with \(w\).
This process looks like:
  • Identify the function: \(f(x) = 3x^2 - x\)
  • Change \(x\) to \(w\): \(f(w) = 3w^2 - w\)
It's that simple! This method works with any variable or number, making analyzing different situations and scenarios easier.
Simplifying Expressions
Simplifying expressions is the process of making them as straightforward as possible. After substitution in a function, simplifying ensures that you have the most concise form of the function.

In our example, we had the function \(f(x) = 3x^2 - x\). After substituting \(w\) for \(x\), we get \(f(w) = 3w^2 - w\).

Here, the expression is already in its simplest form since there are no like terms to combine or further mathematical operations to perform. Simplifying expressions helps us understand, interpret, and use functions more effectively without unnecessary complexity.

Remember: Always double-check each term to confirm whether any further simplification is possible.

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Most popular questions from this chapter

Write the equation of each graph after the indicated transformation\((s)\) The graph of \(y=x^{2}\) is translated thirteen units to the right and six units downward, then reflected in the \(x\) -axis.

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=\left|x^{2}-3\right|$$

Solve each inequality by graphing an appropriate function. State the solution set using interval notation. $$\sqrt{25-x^{2}}>0$$

Solve each problem. The cost in dollars of shipping a machine is given by the function \(C=200+37[w / 100]\) for \(w>0\) where \(w\) is the weight of the machine in pounds. For which values of \(w\) is the cost less than \(\$ 862 ?\)

Use transformations to graph each function and state the domain and range. $$y=-4 x+200$$
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