Problem 103 - Precalculus : Building Concepts and Connections

Question

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Find an expression for \((f \circ f)(t)\), and give the domain of \(f \circ f\)

Step-by-Step Solution

Answer

\((f \circ f)(t) = 9t + 4\), Domain: All real numbers.

Step 1: Finding (f ∘ f)(t) - Substitute t with f(t)

Since \((f \circ f)(t)\) means \(f(f(t))\), substitute \(t\) in \(f(t)=3 t+1\) with \(f(t)\). Thus, we get \(f(f(t)) = 3(f(t)) + 1\)

Step 2: Simplifying (f ∘ f)(t)

Substitute \(f(t) = 3t + 1\) from step 1 to get: \(f(f(t)) = 3(3t + 1) + 1 = 9t + 3 + 1 = 9t + 4\). So, \((f \circ f)(t) = 9t + 4\)

Step 3: Domain of (f ∘ f)(t)

The domain of \((f \circ f)(t)\) is the set of all \(t\) that makes \((f \circ f)(t)\) defined. Given that \((f \circ f)(t) = 9t + 4\) and there is no denominator or square root, this function is valid for all real numbers, \(t\).

Key Concepts

Domain of Function

The domain of a function is the set of input values (usually "x" or "t") for which the function is defined and yields real number outputs. For functions like polynomials, the domain is typically all real numbers, unless there are restrictions like division by zero or square roots of negative numbers.
In our original exercise,

• The function \(f(t) = 3t + 1\) is a linear function. Linear functions have simple domains:
• They are continuous and defined for all real numbers.
• Similarly, when we compose the function with itself to get \((f \circ f)(t) = 9t + 4\), the resulting function is also linear and, therefore, has the same domain.

There's no denominator that could be zero or square roots that require care, so the domain of \(f \circ f\) is all real numbers \(t\). Understanding the domain is crucial as it tells us what inputs can be used without breaking the function's rules.

Function Operations

Function operations involve actions like adding, subtracting, multiplying, dividing, or composing functions together to create new functions.

• Composition, noted as \((f \circ g)(x)\), means plugging one function into another.
• It requires us to evaluate the inner function first, and then use its output as the input to the outer function.

Compose the function \(f\) with itself to form \((f \circ f)(t)\), which means finding \(f(f(t))\). This step involves substituting the expression \(3t + 1\) (from \(f(t)\)) into \(f(t)\) again:

• First, calculate: \(f(t) = 3t + 1\)
• Then substitute back into itself: \(f(f(t)) = 3(3t + 1) + 1 = 9t + 4\)

This operation transforms a simple function into another function, allowing us to predict new outputs for different inputs. Understanding how to perform these operations is key to mastering function manipulation in algebra.

Real Numbers

Real numbers are all the numbers on the number line, including all the rational numbers (like fractions, such as 1/2, or integers like -3) and the irrational numbers (like \(\sqrt{2}\) or \(\pi\)). They encompass everything we commonly use in mathematics, except for imaginary numbers.

• When we talk about the domain of a function, we often refer to all real numbers because functions are generally defined over the sets of real numbers.
• For instance, the real numbers provide the basis for the domain of the function \( (f \circ f)(t) = 9t + 4\). This is because linear functions like this one can take any real number as input and still produce a real number as output.

Understanding real numbers is fundamental as it helps ensure that the operations performed on functions result in valid outputs.