Problem 99 - Precalculus : Building Concepts and Connections

Question

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Find an expression for \((f \circ f)(t),\) and give the domain of \(f \circ f\).

Step-by-Step Solution

Answer

The expression for \(f \circ f(t)\) is \(-t^4\), and the domain of \(f \circ f\) is all real numbers (\(-\infty < t < \infty\)).

Step 1: Find \(f(t)\)

In the given function \(f(t) = -t^2\), replace \(t\) with \(-t^2\). This gives \(f(f(t)) = -(-t^2)^2\).

Step 2: Simplify the equation

Simplify the equation to its simplest form. The inner square will remove the negative sign before \(t^2\), resulting in the equation following this simplification to be \(f(f(t)) = -t^4\).

Step 3: Find the domain of \(f \circ f\)

The function is valid for all real numbers because the fourth power of any real number is also a real number. Therefore, the domain of \(f \circ f\) is all real numbers or \(-\infty < t < \infty\).

Key Concepts

Understanding Function Composition

Function composition involves combining two functions to create a new one. This is like feeding the output of one function directly into another function. For example, if we have two functions, \( f \) and \( g \), the composition \( (f \circ g)(x) \) can be thought of as \( f(g(x)) \). It means we first apply function \( g \) to \( x \) and then apply function \( f \) to the result of \( g(x) \).
In the given exercise, the task is to find \( (f \circ f)(t) \), which means we apply function \( f \) to the output of another \( f \), or \( f(f(t)) \). This helps in understanding how values are transformed through successive mathematical operations. Keep in mind, order matters in function composition! You must apply the functions in the sequence they are mentioned.

Exploring the Domain of a Function

The domain of a function refers to all the possible input values that the function can accept without leading to any undefined situations. For instance, when considering a simple function \( f(x) = x^2 \), any real number can be squared, so the domain of this function can be all real numbers.
For the function composition \( f(f(t)) \) from the exercise, the inner function \( f(t) = -t^2 \) is a polynomial, and polynomials are generally defined for all real numbers. Since exponentiation (raising a number to a positive power) and negation do not impose additional limitations, the composition function \( f(f(t)) = -t^4 \) also accepts any real number as input.
Therefore, the domain of the function \( f \circ f \) remains all real numbers. It's indicated by the interval notation \((-\infty, \infty)\), representing that there are no restrictions on the values that \( t \) can take.

Concept of Real Numbers

Real numbers constitute a vast set that includes all rational and irrational numbers. In simple terms, any number that you can think of on the number line is a real number. This includes integers, fractions, decimals, and special numbers like \( \pi \) and \( \sqrt{2} \).
Real numbers are essential in everyday mathematics and calculus because they provide a comprehensive number system to deal with most mathematical operations. They include:

• Whole numbers, like 0, 1, 2...
• Fractions, like \( \frac{1}{2}, \frac{3}{4} \)
• Decimals, like 3.14 or 2.71828
• Irrational numbers, like \( \sqrt{2} \) and \( \pi \)

When dealing with functions, especially those with polynomials like in our exercise involving function composition, considering real numbers is crucial because these functions are typically continuous and defined across all real numbers. Thus, understanding real numbers helps grasp the domain of these functions effectively.