Question

\(f(x)=x^4+1, g(x)=f(x+2)\)

Short answer

The transformed function, \(g(x)\), is \(g(x) = (x+2)^4 + 1\).

Step-by-step solution

Identify the Functions

In this problem, there are two functions provided. The given functions are \(f(x) = x^4 + 1\) and \(g(x) = f(x+2)\). The goal is to find the form of function \(g(x)\) by replacing \(x\) in function \(f(x)\) with the expression \(x+2\).

Substitute the Expression for x in \(f(x)\)

The function \(g(x)\) is defined as \(f(x+2)\). This means we substitute within \(f(x)\) every \(x\) with \(x+2\). So, \(g(x) = f(x+2)\) becomes \(g(x) = (x+2)^4 + 1\).

Simplify the Function \(g(x)\)

The function, \(g(x) = (x+2)^4 + 1\), is already a simplified form. Thus, this constitutes the final step.

Key concepts

Understanding Polynomial Functions

Polynomial functions are a type of algebraic expression that consists of variables raised to non-negative integer powers combined with coefficients. These functions operate under addition, subtraction, and multiplication. For example, the function \( f(x) = x^4 + 1 \) is a polynomial function of degree 4. The degree is determined by the highest power of the variable present, which in this case is 4. Polynomial functions are fundamental in various fields of science and mathematics because they model a wide range of phenomena. They are continuous, smooth, and have simple derivatives, making them manageable for calculations and practical for real-world applications. When dealing with polynomial functions, remember that the degree of the polynomial indicates the number of roots or solutions it might have, although not all solutions have to be real values.

Decoding Function Notation

Function notation is a way to label functions clearly and efficiently. It helps to express a mathematical relationship between variables without ambiguity. For instance, in the function \( f(x) = x^4 + 1 \), \( f \) is the function name, \( x \) is the variable or input, and \( x^4 + 1 \) is the output or the expression defining the function.Using function notation makes it easy to evaluate functions at different points. For example, \( f(2) \) tells us to substitute \( x \) with 2 in the function \( f(x) \). This systematic approach keeps our calculations organized and clear.Furthermore, function notation aids in the composition of functions, where one function is applied within another, as seen in the problem with \( g(x) = f(x+2) \). Function notation helps streamline these operations and reduces potential errors.

Mastering Expression Substitution

Expression substitution involves replacing one part of an equation or function with another expression or value. This is a fundamental technique used in algebra to simplify or evaluate functions.In the exercise, substituting involves taking the expression \( x+2 \) and replacing \( x \) in the given polynomial function \( f(x) = x^4 + 1 \) to form \( g(x) = f(x+2) \). This translates to \( g(x) = (x+2)^4 + 1 \).Executing this method correctly is essential when working with nested functions or composing multiple functions into one. By substituting correctly, you can simplify complex expressions and solve problems more easily. Remember, always pay attention to substitution steps as they can drastically alter the function’s expression and the result of calculations.