Question

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f-g)(2 t)\)

Short answer

The result of \((f-g)(2t)\) is \(4t -4t^2 + 3\).

Step-by-step solution

Evaluate \(f(2t)\)

Replace every instance of x in the definition of f(x) with 2t. In this case, this means replacing x in \(f(x) = 2x + 1\) with 2t: \(f(2t) = 2*(2t) + 1 = 4t + 1\).

Evaluate \(g(2t)\)

Replace every instance of x in the definition of g(x) with 2t. In this case, this means replacing x in \(g(x) = x^2 - 2\) with 2t: \(g(2t) = (2t)^2 - 2 = 4t^2 - 2\).

Subtract \(g(2t)\) from \(f(2t)\)

To compute \(f(2t) - g(2t)\), use the values obtained from Steps 1 and 2 and subtract \(g(2t)\) from \(f(2t)\): \((f-g)(2t) = f(2t) - g(2t) = (4t+1) - (4t^2 - 2) = 4t + 1 - 4t^2 + 2 = 4t -4t^2 + 3\).

Key concepts

Polynomial Functions

In algebra, polynomial functions are expressions that involve variables raised to whole number exponents, which are combined using addition, subtraction, and multiplication. Each term in a polynomial has a coefficient, which is a number that multiplies the variable. For example, in the function \( f(x) = 2x + 1 \), "2x" is a term where "2" is the coefficient of \( x \), and "1" is a constant term. Polynomials are named based on their degree and number of terms, such as linear, quadratic, or cubic polynomials.
Linear polynomials like \( f(x) = 2x + 1 \) have the degree of one, meaning the highest power of \( x \) is one. The simplest operations for polynomials include addition, subtraction, and multiplication.
Understanding the structure of polynomials allows for easier manipulation and solving of algebraic problems, particularly in function evaluation, when specific variable values are substituted into the polynomial to compute an output.

Function Operations

Function operations involve combining functions through addition, subtraction, multiplication, and division, where applicable. When dealing with operations such as \((f-g)(x)\), it requires determining the expression by substituting specific values into the functions \(f(x)\) and \(g(x)\).
For instance, in the given problem, we assess \((f-g)(2t)\). First, substitute \(2t\) into each function separately to find \(f(2t)\) and \(g(2t)\). Then, subtract the result of \(g(2t)\) from \(f(2t)\).
This process requires careful attention to order of operations and like-terms, particularly when dealing with polynomial functions. This way, it becomes easier to simplify expressions and carry out algebraic calculations accurately.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations. In the context of function evaluation and operations, such expressions often involve substituting values for variables and simplifying the results. For example, the function \( f(x) = 2x + 1 \) becomes an algebraic expression \( 4t + 1 \) when \( 2t \) is substituted for \( x \).
Algebraic expressions are vital in solving equations and evaluating functions because they allow for the representation of complex relationships in a simplified and manipulable form. When managing expressions like \(f(x) - g(x)\), recognize that each term must be dealt with individually, ensuring the correct execution of mathematical operations.

• Look out for correct application of arithmetic operations.
• Pay attention to combining like terms.
• Always simplify expressions to their simplest form.

Mastery of handling algebraic expressions smoothly is key to succeeding in advanced mathematics courses.