Problem 21 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((f - g)(3t)\)

Step-by-Step Solution

Answer

The result of the function \((f - g)(3t)\) is \(9t^{2} - 3t + 5\).

Step 1: Substitute the provided variable in the functions

First, substitute \(x = 3t\) in both functions f and g, getting \(f(3t) = (3t)^2 + 1\) and \(g(3t) = 3t - 4\).

Step 2: Simplify the functions

Simplify both of the obtained functions which yields \(f(3t) = 9t^2 + 1\) and \(g(3t) = 3t - 4\).

Step 3: Calculate the result of the expression

Lastly, we compute the difference \((f - g)(3t) = f(3t) - g(3t) = (9t^2 + 1) - (3t - 4)\), which simplifies to \(9t^{2} - 3t + 5\) as the output.

Key Concepts

Difference of Functions

Understanding the concept of the difference of functions is essential in algebra and calculus. Essentially, when we talk about the difference of two functions, we're referring to a new function that is created by subtracting one function from another. For example, if we have two functions, say, f(x) and g(x), their difference (f - g)(x) is a new function obtained by subtracting g(x) from f(x).

Mathematically,

(f - g)(x) = f(x) - g(x).

So, if students are asked to find (f - g)(3t) for functions f(x) = x^2 + 1 and g(x) = x - 4, they need to evaluate both functions at x = 3t before subtracting. This reinforces the concept that operations on functions, such as differences, can be performed pointwise at specific inputs.

Function Evaluation

Evaluating a function means finding the output value of a function for a particular input. This is a fundamental skill in mathematics that allows students to work with functions effectively. To evaluate a function like f(3t), we replace the variable x with 3t in the function's formula, and then simplify the expression.

For the polynomial function f(x) = x^2 + 1, evaluating f(3t) would involve squaring 3t to get (3t)^2, which gives us 9t^2. Then, we add 1 to complete the evaluation:

f(3t) = 9t^2 + 1

Evaluating functions is about following the prescribed operations within the function's definition for any given input.

Polynomial Functions

Polynomial functions are a class of functions that are built from sums of powers of the variable, with each term including a coefficient. Our example functions, f(x) and g(x), are both polynomial functions. f(x) = x^2 + 1 is a quadratic polynomial, since its highest power of x is 2, and g(x) = x - 4 is a linear polynomial, as its highest power of x is 1.

These functions are important because they are relatively simple to evaluate, differentiate, and integrate, making them foundational for more complex mathematical concepts. Moreover, polynomial functions often model real-world phenomena, such as projectile motion (quadratic polynomials) or cost functions (often linear polynomials).

Simplifying Expressions

Simplifying expressions in algebra is a critical skill that allows for clearer understanding and easier computation of functions. The process involves reducing an expression to its most basic form without changing its value. This may include combining like terms, factoring, canceling terms, and applying exponent rules.

As seen in the example, to simplify the expression

(f - g)(3t) = (9t^2 + 1) - (3t - 4)

you perform operations such as distributing negative signs and combining like terms. The simplified result is

9t^2 - 3t + 5

Simplifying expressions aids in spotting patterns, solving equations, and understanding the behavior of functions more profoundly.