Problem 39 - Contemporary Precalculus

Question

Expand and simplify the rule of the function \(g .\) $$f(x)=x^{3}-2 x^{2}+x-4 \text { and } g(x)=f(x+1)$$

Step-by-Step Solution

Answer

Based on the step-by-step solution provided: Question: Expand and simplify the function `g(x) = f(x+1)` when `f(x) = x^3 - 2x^2 + x - 4`. Answer: g(x) = x^3 + x^2 - 4

Step 1: Understand the given functions

We are given two functions, f(x) and g(x). The function f(x) is defined as \(f(x) = x^3 - 2x^2 + x - 4\). The function g(x) is defined as the function f evaluated at \(x+1\), meaning that \(g(x) = f(x+1)\).

Step 2: Substitute x+1 into f(x)

We will substitute x+1 into f(x) to find the expression of g(x). Replace x in the expression of f(x) with x+1: $$ g(x) = f(x+1) = ((x+1)^3 - 2(x+1)^2 + (x+1) - 4) $$

Step 3: Expand and simplify the expression

Now we will expand and simplify the expression we obtained in step 2: $$ g(x) = ((x+1)^3 - 2(x+1)^2 + (x+1) - 4) \\ = ((x^3+3x^2+3x+1) - 2(x^2+2x+1) + x + 1 - 4) \\ = (x^3+3x^2+3x+1 - 2x^2-4x-2 + x + 1 - 4) \\ = (x^3+3x^2+3x+1 - 2x^2-4x-2 + x + 1 - 4) \\ =x^3 + (3-2)x^2 + (3-4+1)x + (1-2+1-4) \\ =x^3 + x^2 - 4 $$ So, the rule for g(x) is simplified as: $$ g(x) = x^3 + x^2 - 4 $$

Key Concepts

Polynomial Functions

Polynomial functions are an essential building block of algebra that represent expressions composed of variables raised to power and their coefficients. These functions are formed by adding together terms like \(x^3\), \(-2x^2\), \(x\), and constants such as \(-4\). The highest exponent in the polynomial dictates its degree, which informs us about the potential number of roots and the shape of its graph. In the given exercise, the original function \(f(x) = x^3 - 2x^2 + x - 4\) is a degree 3 polynomial.

• Degrees: Indicate the highest power of the variable (e.g., cubic means the highest degree is 3).
• Coefficients: The numbers multiplying each term (e.g., -2 in \(-2x^2\)).
• Constant Term: A term without a variable (e.g., \(-4\) in the given function).

Understanding polynomial functions helps in expanding, simplifying, and transforming them. Each manipulation can affect their properties, like roots or intercepts, which are crucial for graphing and solving equations.

Function Expansion

Function expansion involves extending a polynomial by substituting variables or values, leading to a new polynomial expression. In the exercise, the transformation from \(f(x)\) to \(g(x) = f(x+1)\) demanded expanding the function \(f(x)\) by replacing \(x\) with \(x+1\). This method broadens the function, affecting each term as shown:

• Replace \(x\) with a new expression \((x+1)\) in every term of \(f(x)\).
• Calculate each power post-substitution, e.g., \((x+1)^3\) becomes \(x^3 + 3x^2 + 3x + 1\).
• Adjust and combine results into a comprehensive polynomial.

The expansion process is vital, as it prepares the function for simplification, while ensuring the integrity and original form of the function transform correctly.

Simplified Expressions

Simplified expressions aim to reduce a complex polynomial into its most concise form while retaining the same value. In our exercise's context, after expanding \(g(x) = f(x+1)\), we simplified the function. Simplification entails collecting like terms and performing algebraic operations meticulously. Here is how the simplification process unfolded:

• Combine like terms: This means adding or subtracting coefficients of terms with the same degree, e.g., \((3 - 2)x^2\) gets simplified to just \(x^2\).
• Follow the order of operations: Carefully compute addition, subtraction, and any multiplicative conjunctions.
• Check and recheck the work: Ensure that each step aligns correctly, leading to the ultimate clean expression.

Through simplification, the function \(g(x) = x^3 + x^2 - 4\) represents the clean, polynomial form of its expanded version. Simplified expressions are crucial for interpreting the characteristics and behaviors of functions efficiently.