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Chapter 0: Problem 94

Evaluate \(f(4)\). \(f(x)=x^{-1 / 2}\)

Short Answer

Expert verified
The value of \( f(4) \) is \( \frac{1}{2} \).

Step by step solution

01

Understand the Function

The function given is a power function: \( f(x) = x^{-1/2} \). This means for any value of \(x\), you will raise \(x\) to the power of \(-1/2\).
02

Substitute the Given Value

Substitute \(x = 4\) into the function: \[ f(4) = 4^{-1/2} \]
03

Simplify the Exponent

Recall that \(a^{-b}\) can be written as \( \frac{1}{a^b} \). Thus: \[ 4^{-1/2} = \frac{1}{4^{1/2}} \]
04

Evaluate the Expression

Simplify \( 4^{1/2} \). Since \( 4^{1/2} = \sqrt{4} = 2 \), we get: \[ \frac{1}{4^{1/2}} = \frac{1}{2} \]
05

Write the Final Answer

Thus, \( f(4) = \frac{1}{2} \).

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Key Concepts

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

Power Functions
A power function is any function of the form \(f(x) = kx^n\), where \(k\) and \(n\) are constants. Power functions can have different characteristics depending on the value of \(n\). For example:
  • If \(n\) is positive and greater than or equal to 1, the function is a polynomial.
  • If \(n\) is negative, it represents a reciprocal relationship.
  • If \(n\) is a fraction, it represents roots.
When \(n\) is \(-1/2\), as in the given function \(f(x) = x^{-1/2}\), it means we are dealing with a reciprocal of a square root. This is key when working on evaluating the given function.
Exponents
Exponents are a fundamental concept in algebra. They indicate how many times a number, called the base, is multiplied by itself. Here are a few basic rules:
  • \(a^m \times a^n = a^{(m+n)}\)
  • \(a^{-n} = \frac{1}{a^n}\)
  • \(a^{1/n} = \sqrt[n]{a}\)
  • \(a^0 = 1\) (for any non-zero \(a\))
In this exercise, we used the rule \(a^{-b} = \frac{1}{a^b}\) to simplify the negative exponent. For \(4^{-1/2}\), this becomes \(\frac{1}{4^{1/2}}\), which leads to taking the square root of 4.
Simplifying Expressions
Simplifying expressions often involves several algebraic techniques, including combining like terms, using the distributive property, and applying exponent rules. When simplifying the expression \(4^{-1/2}\), we used the fact that \(a^{-b}\) can be transformed into \(\frac{1}{a^b}\), changing an exponentiation problem into a fraction. Next, to simplify the expression further, we evaluated \(4^{1/2}\), recognizing that finding the square root of 4 reduces our problem to \(\frac{1}{2}\). Each step follows logically, anchoring on these simplification rules.
Square Roots
A square root of a number \(a\) is a value \(b\) such that \(b^2 = a\). Notationally, this is expressed as \(\sqrt{a}\). Key points include:
  • The square root of a positive number \(n\) has two values: \(+\sqrt{n}\) and \(-\sqrt{n}\).
  • The square root of 4 is written as \(\sqrt{4}\) and equals 2, since \(2^2 = 4\).
  • It is important to recognize when simplifying that the square root operation undoes the squaring of a number.
In the function \(f(x) = x^{-1/2}\), after transforming it into \(\frac{1}{\sqrt{4}}\), we found \(\sqrt{4} = 2\). This allows us to evaluate the expression \(\frac{1}{2}\) easily.

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