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Chapter 10: Problem 103

Determine the domain of each function described. $$ f(t)=7+\sqrt[8]{t^{8}} $$

Short Answer

Expert verified
The domain is all real numbers.

Step by step solution

01

Understand the Function

The function given is \( f(t) = 7 + \sqrt[8]{t^8} \). Analyze each part to determine potential restrictions on the domain.
02

Simplify the Function

Simplify \( \sqrt[8]{t^8} \). Since both the radicand and the index are 8, it simplifies to \( \sqrt[8]{t^8} = t \). Therefore, the function is now \( f(t) = 7 + t \).
03

Identify Restrictions

There are no constraints like division by zero or taking the even root of a negative number in the simplified function \( 7 + t \). Therefore, there are no restrictions on the domain.
04

Determine the Domain

Since there are no restrictions, the domain includes all real numbers.

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Key Concepts

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

function analysis
To find the domain of a function, we start by analyzing its components. The function given here is: \( f(t) = 7 + \sqrt[8]{t^8} \) In function analysis, we look at particular areas worth noting, such as exponents (as in our example with the root), fractions, and any other elements that might restrict the domain. In this case, we need to see whether the eighth root of \( t^8 \) imposes any restrictions.
simplifying functions
Simplifying a function involves reducing it to its most basic form. This can make it easier to understand and analyze. For the function \( f(t) = 7 + \sqrt[8]{t^8} \), we see that the eighth root of \( t^8 \) simplifies straight to \( t \). This means that our function becomes: \( f(t) = 7 + t \) Simplifying functions can often make apparent any restrictions or constraints that were not obvious before.
domain restrictions
When determining the domain of a function, it's essential to identify potential restrictions. These can be due to:
  • Division by zero
  • Even roots of negative numbers
  • Logarithms of non-positive numbers
In our example, since the simplified function is \( 7 + t \), there are no fractions or even roots. Because of this, there are no restrictions on \( t \). Any number can be substituted for \( t \) without leading to invalid results.
real numbers
The domain of a function consists of all possible input values (t in our case) that will give valid outputs. For the function \( f(t) = 7 + t \), since there are no restrictions, the domain is simply all real numbers. Real numbers include every number you can think of: positive, negative, zero, integers, fractions, and irrational numbers. In mathematical notation, we write this domain as: \( (-\infty, \infty) \) This shows that the function is defined for every real number value of t.

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