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Chapter 0: Problem 91

Evaluate \(f(4)\). \(f(x)=x^{-1}\)

Short Answer

Expert verified
f(4) = \(\frac{1}{4}\)

Step by step solution

01

Understand the function

The given function is defined as follows: \(f(x) = x^{-1}\).This means that for any input value of \(x\), the output value of the function is the reciprocal of \(x\).
02

Substitute the value

To evaluate \(f(4)\), substitute \(4\) for \(x\) in the function:\(f(4) = 4^{-1}\).
03

Simplify the expression

Now, simplify the expression \(4^{-1}\):\(4^{-1}\) is the reciprocal of \(4\), which means\(f(4) = \frac{1}{4}\).

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Key Concepts

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

Function Evaluation
Function evaluation is simply the process of finding the output of a function given a specific input. In our exercise, we're asked to evaluate the function at a particular point. Given the function notation, which is typically written as \(f(x)\), it means we need to substitute the given value into the function and perform the necessary calculations to find the result.
The steps are:
  • Identify the input value.
  • Substitute the input into the function.
  • Perform the arithmetic operations to obtain the output.
For example, if you're asked to evaluate \(f(4)\) for \(f(x)=x^{-1}\), you substitute 4 for x and simplify the expression.
Reciprocal
The reciprocal of a number is defined as one divided by that number. For a given number \(x\), the reciprocal is denoted as \( x^{-1} \). The reciprocal has some important properties:
  • You can find the reciprocal of any non-zero number. For example, the reciprocal of \(4\) is \( \frac{1}{4} \).
  • The reciprocal of the reciprocal of a number is the number itself. For instance, the reciprocal of \( \frac{1}{4} \) is \(4\).
  • Multiplying a number by its reciprocal always equals 1. For example, \(4 \times \frac{1}{4} = 1\).
In our exercise, the function \(f(x) = x^{-1} \) tells us to find the reciprocal of the input \(x\). So, for \(f(4)\), we find the reciprocal of \(4\), which is \(\frac{1}{4}\).
Exponents
Exponents are a mathematical notation indicating the number of times a number, called the base, is multiplied by itself. In our function, \(x^{-1}\), the exponent \(-1\) tells us something important. Here's what you need to know:
  • A positive exponent \(n\) means multiplying the base by itself \(n\) times, such as \(2^3 = 2 \times 2 \times 2 = 8\).
  • A negative exponent \(n\) means taking the reciprocal of the base raised to \(n\). For example, \(4^{-1} = \frac{1}{4}\).
  • Exponents follow specific rules, like the product rule \(a^m \times a^n = a^{m+n}\) and the power rule \((a^m)^n = a^{m \times n}\).
In our exercise, \(x^{-1}\) means that we're taking the reciprocal of \(x\), which translates to taking one divided by \(x\). For example, four raised to the power of negative one yields \(\frac{1}{4}\).

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Most popular questions from this chapter

In Exercises , find the zeros of the function. (Use the specified viewing window.) $$ f(x)=\frac{x}{x+2}-x^{2}+1 ;[-1.5,2] \text { by }[-2,3] $$

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