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Chapter 0: Problem 96

Evaluate \(f(4)\). \(f(x)=x^{0}\)

Short Answer

Expert verified
f(4) = 1

Step by step solution

01

Understand the function

The given function is simplified as follows. Recall that any non-zero number raised to the power of zero is equal to one. The function given is: \[ f(x) = x^{0} \]
02

Substitute the value

To find \( f(4) \), substitute \( x = 4 \) into the function: \[ f(4) = 4^{0} \]
03

Evaluate the expression

Evaluate the expression using the rule that any non-zero number raised to the zero power is 1: \[ 4^{0} = 1 \]
04

Conclusion

Therefore, the value of \( f(4) \) is 1.

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Key Concepts

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

Exponent Rules
Exponent rules are essential to understand how to simplify expressions involving powers. One critical rule is the Zero Exponent Rule. This rule states that any non-zero number raised to the power of zero is equal to one. It's written mathematically as: \[a^0 = 1\] where \(a\) is any non-zero number. This rule helps simplify functions and expressions quickly. Remember: it's only valid for non-zero bases. Let's look at an example to visualize:
Suppose we have \(7^0\). According to the Zero Exponent Rule, \(7^0 = 1\). This rule is crucial for evaluating functions like \(f(x) = x^0\). Once you master exponent rules, simplifying expressions becomes much easier!
Function Evaluation
Function evaluation is the process of determining the output of a function for a particular input. Functions are often written in the form \[f(x)\], where \(x\) is the input variable. To evaluate a function, follow these simple steps:
  • Identify the function and the given input.
  • Substitute the input value into the function.
  • Simplify the resulting expression.
For example, consider the function \(f(x) = x^0\). If we want to evaluate \(f(4)\), we substitute \(4\) for \(x\), resulting in \(4^0\). Using the Zero Exponent Rule, we can simplify this to \(1\). Therefore, \(f(4) = 1\). Learning function evaluation is pivotal for understanding more complex mathematical concepts.
Zero Exponent
The Zero Exponent is a special case in exponent rules where any non-zero number raised to the power of zero equals one. This concept is incredibly useful in simplifying seemingly complex expressions. The mathematical form is: \[a^0 = 1\] for any non-zero \(a\). Here's how you can remember it:
  • Any number to the zero power simplifies to one.
  • The base must be non-zero for this rule to apply.
  • It's a shortcut that saves time in calculations.
Let's apply this to a function like \(f(x) = x^0\). When evaluating at \(x = 4\), the expression becomes \(4^0\), which simplifies to \(1\). Thus, \(f(4) = 1\). Understanding Zero Exponent will give you the power to simplify many mathematical expressions effortlessly.

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Most popular questions from this chapter

Convert the numbers from graphing calculator form to standard form (that is, without E). 8.103E-4

In Exercises \(53-56\), compute \(f(1), f(2)\), and \(f(3)\). $$ f(x)=\left\\{\begin{array}{ll} \sqrt{x} & \text { for } 0 \leq x<2 \\ 1+x & \text { for } 2 \leq x \leq 5 \end{array}\right. $$

$$ \begin{array}{l} \text { Find the points of intersection of the pairs of curves in Exercises }\\\ \end{array} $$ $$ y=3 x^{2}+9, y=2 x^{2}-5 x+3 $$

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Draw the following intervals on the number line. $$ [-1,4] $$
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