Problem 18 - Contemporary Precalculus

Question

Find the rules of the composite functions \(f \circ g\) and \(g \circ f\). $$f(t)=\cos ^{2}(t-2) ; \quad g(t)=5 t+2$$

Step-by-Step Solution

Answer

Question: Find the composite functions \(f \circ g(t)\) and \(g \circ f(t)\) for the given functions \(f(t) = \cos^2(t - 2)\) and \(g(t) = 5t + 2\). Answer: The composite functions are: 1. \(f \circ g(t) = \cos^2(5t)\) 2. \(g \circ f(t) = 5(\cos^2(t - 2)) + 2\).

Step 1: Compute \(f \circ g(t)\)

To compute \(f \circ g(t)\), we substitute the expression for \(g(t)\) into the function \(f(t)\): $$f \circ g(t) = f(g(t)) = \cos^2(g(t) - 2)$$ Now substitute the expression for \(g(t)\): $$f \circ g(t) = \cos^2((5t + 2) - 2)$$

Step 2: Simplify the expression for \(f \circ g(t)\)

Next, we simplify the expression inside the cosine function: $$f \circ g(t) = \cos^2(5t)$$

Step 3: Compute \(g \circ f(t)\)

Now, to compute \(g \circ f(t)\), we substitute the expression for \(f(t)\) into the function \(g(t)\): $$g \circ f(t) = g(f(t)) = 5f(t) + 2$$ Now substitute the expression for \(f(t)\): $$g \circ f(t) = 5(\cos^2(t - 2)) + 2$$

Step 4: Final answer

The composite functions are: $$f \circ g(t) = \cos^2(5t)$$ $$g \circ f(t) = 5(\cos^2(t - 2)) + 2$$

Key Concepts

Function Composition

Function composition is a powerful tool often used in mathematics, especially when dealing with multiple functions. It involves applying one function to the results of another function. Imagine you have two functions, say \(f(t)\) and \(g(t)\). When you compose these functions, you create a new function by inserting \(g(t)\) into \(f(t)\), which is noted as \(f \circ g(t)\).

Here's a simple analogy: think of it like cooking a dish where \(g(t)\) represents preparing an ingredient, and \(f(t)\) represents adding that ingredient to the final dish. By first completing \(g(t)\) and then inserting it into \(f(t)\), you produce the composite function \(f \circ g(t)\). This principle can be reversed, too, yielding \(g \circ f(t)\), which means applying \(g(t)\) after \(f(t)\).

• To find \(f \circ g(t)\), substitute \(g(t)\) into \(f(t)\)
• To find \(g \circ f(t)\), substitute \(f(t)\) into \(g(t)\)

Trigonometric Functions

Trigonometric functions are mathematical functions often encountered in precalculus and calculus, crucial for studying angles and periodic phenomena. One very common trigonometric function is the cosine function, noted as \(\cos(x)\).

The exercise provided here deals primarily with the cosine function, raised to the power of two, written as \(\cos^2(x)\). This is shorthand for \((\cos(x))^2\). These functions often arise in the study of waves and oscillations, as well as in the properties of triangles.

When working with composite functions involving trigonometric expressions like \(\cos(x)\), it’s helpful to simplify the expressions as much as possible. For instance: if you have an expression \(\cos(g(t) - 2)\), simplifying the inner part \(g(t) - 2\) first makes it easier to evaluate or further manipulate the function.

Precalculus

Precalculus is an essential field of mathematics that prepares students for calculus by covering fundamental concepts such as functions, trigonometry, and complex numbers. It sets the stage for more advanced topics in mathematics.

In precalculus, one of the critical skills is understanding and manipulating functions, as seen in the concept of function composition. Precalculus students encounter trigonometric functions like sine, cosine, and tangent, as well as their transformations and uses.

• Key skills in precalculus include graphing functions and solving equations involving functions like \(\cos(x)\)
• Understanding compositions helps in modeling and solving real-world problems

Mastering these concepts through exercises, such as calculating and simplifying composite functions, is vital for easing the transition into calculus.