Problem 62
Question
Find the domain of each function. $$ f(t)=\frac{5}{t^{2}+9} $$
Step-by-Step Solution
Verified Answer
The domain of the function \( f(t)=\frac{5}{t^{2}+9} \) is all real numbers.
1Step 1: Identify the Denominator
The denominator of the function \( f(t)=\frac{5}{t^{2}+9} \) is \( t^{2}+9 \).
2Step 2: Solve the denominator equation
We set the denominator equal to zero and solve for 't' to find which values would be not valid for the function. Here's how: Set up the equation \( t^{2}+9=0 \). This equation has no real solutions since the smallest value of \( t^{2} \) is 0, and even then, the equation would be \(0+9= 0\) which is not true. Therefore there are no 't' values which will make the denominator equal to zero.
3Step 3: Find the Domain
Since there are no values of 't' that will lead to division by zero, the domain of function \( f(t) \) is all real numbers.
Key Concepts
Rational FunctionsDenominatorDivision by Zero
Rational Functions
Rational functions are expressions that appear as the quotient of two polynomials. These functions are defined as follows: a rational function is any function that can be expressed as the ratio \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
Why does \( Q(x) eq 0 \)? Because division by zero is undefined in mathematics, meaning it does not produce a valid or meaningful result.
Hence, a crucial part of understanding rational functions is determining their domain. This involves finding out when the denominator, \( Q(x) \), becomes zero, which helps identify the "forbidden" values of \( x \).
For some rational functions, the solutions might be real numbers, while for others, like the exercise given, there might be no real zeroes at all.
Why does \( Q(x) eq 0 \)? Because division by zero is undefined in mathematics, meaning it does not produce a valid or meaningful result.
Hence, a crucial part of understanding rational functions is determining their domain. This involves finding out when the denominator, \( Q(x) \), becomes zero, which helps identify the "forbidden" values of \( x \).
For some rational functions, the solutions might be real numbers, while for others, like the exercise given, there might be no real zeroes at all.
Denominator
In the context of rational functions, the denominator is the polynomial that appears in the bottom part of a fraction. It holds critical information about the function's behavior and constraints. A rational expression, \( \frac{5}{t^2 + 9} \), has a denominator \( t^2 + 9 \).
To determine when a rational function is undefined, we focus on the values of the variables that make the denominator zero.
For example, if the denominator were \( t^2 - 9 \), we would find the zeros by solving \( t^2 - 9 = 0 \), leading us to the values \( t = 3 \) and \( t = -3 \).
However, in the exercise \( t^2 + 9 = 0 \), there are no real number solutions since the smallest \( t^2 \) can be is 0, making the function well-defined for all real numbers.
To determine when a rational function is undefined, we focus on the values of the variables that make the denominator zero.
For example, if the denominator were \( t^2 - 9 \), we would find the zeros by solving \( t^2 - 9 = 0 \), leading us to the values \( t = 3 \) and \( t = -3 \).
However, in the exercise \( t^2 + 9 = 0 \), there are no real number solutions since the smallest \( t^2 \) can be is 0, making the function well-defined for all real numbers.
Division by Zero
Division by zero is a mathematical operation that is undefined. When the denominator of a rational function equals zero, the operation does not conform to the rules of arithmetic because dividing by zero does not yield any meaningful value.
The importance of verifying that a denominator in any function does not equal zero at any point cannot be stressed enough. In other words, identifying the points where division by zero could occur is key in defining the function's domain. In mathematical terms, it's crucial to solve the equation \( Q(x) = 0 \).
For our function \( f(t) = \frac{5}{t^2 + 9} \), setting the denominator \( t^2 + 9 = 0 \) shows us there are no real values that result in division by zero. Thus, this issue becomes nonexistent for this function, allowing the domain to include all real numbers.
The importance of verifying that a denominator in any function does not equal zero at any point cannot be stressed enough. In other words, identifying the points where division by zero could occur is key in defining the function's domain. In mathematical terms, it's crucial to solve the equation \( Q(x) = 0 \).
For our function \( f(t) = \frac{5}{t^2 + 9} \), setting the denominator \( t^2 + 9 = 0 \) shows us there are no real values that result in division by zero. Thus, this issue becomes nonexistent for this function, allowing the domain to include all real numbers.
Other exercises in this chapter
Problem 61
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