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Chapter 7: Problem 107

Let \(f(x)=3 x-1\) and \(g(x)=\frac{1}{x}\). $$ \text { Find } f\left(\frac{1}{3}\right) $$

Short Answer

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Step by step solution

01

Identify the function

The function is given as \(f(x) = 3x - 1\). We need to find the value of this function at a specific point.
02

Substitute the value into the function

We need to find \(f\left( \frac{1}{3} \right)\). Substitute \(\frac{1}{3}\) in place of \(x\) in the function: \(f\left( \frac{1}{3} \right) = 3 \left( \frac{1}{3} \right) - 1\).
03

Simplify the expression

Perform the multiplication inside the function: \(3 \left( \frac{1}{3} \right) = 1\). Then subtract 1 from the result: \(1 - 1\).
04

Write down the final answer

The result of the subtraction is \(0\), so \(f\left( \frac{1}{3} \right) = 0\).

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Key Concepts

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

Function Notation
Understanding function notation is crucial in mathematics. Function notation is a way to represent functions using symbols. A function like \(f(x)\) means we have a function named \(f\), and \(x\) is the input variable. This notation tells us that every time we input a different value for \(x\), the function \(f\) will produce a corresponding output. For example, in the function \(f(x) = 3x - 1\), if you plug in \(x = 2\), you would write \(f(2)\) and calculate to find the output.
Substitution
Substitution is the process of replacing a variable in an equation with a specific value. In our exercise, we needed to find the value of the function \(f\) at \(x = \frac{1}{3}\).
To do this:
  • Locate the function we need, which is \(f(x) = 3x - 1\).
  • Identify that we need to substitute \(\frac{1}{3}\) in place of \(x\).
  • So, we replace every \(x\) in \(f(x)\) with \(\frac{1}{3}\), making it \(f\left( \frac{1}{3} \right) = 3 \left( \frac{1}{3} \right) - 1\).
This substitution allows us to evaluate or simplify the expression next.
Simplification
Simplification involves reducing an expression to its simplest form. After substituting \(\frac{1}{3}\) into the function, we have \(f\left( \frac{1}{3} \right) = 3 \left( \frac{1}{3} \right) - 1\). Next, we perform the arithmetic step-by-step:
  • First, calculate the multiplication: \(3 \left( \frac{1}{3} \right) = 1\).
  • Then, perform the subtraction: \(1 - 1 = 0\).
Thus, we have simplified the expression to find that \(f\left( \frac{1}{3} \right) = 0\). Simplifying expressions is key to finding the final result in function evaluations.

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