Problem 42 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Given the function \(f(x) = 2x^3 - x + 4\), find the rule for the function \(g(x) = f(x+3)\). Answer: The rule for the function \(g(x)\) is \(g(x) = 2x^3 + 18x^2 + 53x + 55\).

Step 1: Substitute \(x+3\) into \(f(x)\)

To find \(g(x)\), we will replace \(x\) in the original function \(f(x)\) with \((x+3)\). This means that wherever we see \(x\) in \(f(x)\), we will replace it with \((x+3)\). $$ g(x) = f(x + 3) = 2(x+3)^3 - (x+3) + 4 $$

Step 2: Expand the cubic term

We will now expand the cubic term \((x+3)^3\). We can use the binomial expansion formula for this: $$ (x+3)^3 = x^3 + 3\cdot(3)x^2 + 3\cdot(3)^2x + 3^3 $$ So, the expression of \(g(x)\) now becomes: $$ g(x) = 2(x^3 + 9x^2 + 27x + 27) - (x+3) + 4 $$

Step 3: Distribute the constant and simplify

Now, we will distribute the constant, merging the terms, and simplifying the expression: $$ g(x) = 2x^3 + 18x^2 + 54x + 54 - x - 3 + 4 $$ Combine like terms: $$ g(x) = 2x^3 + 18x^2 + (54 - 1)x + (54 - 3 + 4) $$ Finally, we obtain the expanded and simplified form of \(g(x)\): $$ g(x) = 2x^3 + 18x^2 + 53x + 55 $$

Key Concepts

Function Expansion

In mathematics, expanding a function involves transforming one expression into an equivalent expression that is more detailed or spread out. In our exercise, we expand the function by substituting the expression

• First, identify the function that needs expansion. Here, we have the function \(f(x) = 2x^3 - x + 4\).
• Next, the function is modified by evaluating it at \(x + 3\) instead of just \(x\). This involves substituting every \(x\) in \(f(x)\) with \(x+3\).
• This substitution results in a new function \(g(x) = f(x+3) = 2(x+3)^3 - (x+3) + 4\).

Now, the function is ready for further expansion and simplification. By changing \(f(x)\) to \(g(x)\), we're preparing to perform more steps which can reveal more properties of the expression. Expanding can make it easier to address and manipulate for different uses, such as in calculus or theorem proving.

Binomial Theorem

The binomial theorem is a powerful method used in algebra to expand expressions raised to a power, such as

• According to the binomial theorem, any power \((a + b)^n\) can be expanded as a sum, often involving combinations, of the terms.
• In our exercise, the component \((x+3)^3\) fits this pattern and can be expanded through the binomial theorem.
• The complete expansion involves the pattern: \[(x+3)^3 = x^3 + 3 \cdot 3x^2 + 3 \cdot 9x + 27\]This method rearranges the simpler term to reveal all components clearly.

Each coefficient and power can be derived from the binomial coefficients, calculated using factorials, although in our case this is simplified by direct multiplication due to constant values 3 and 9. This expansion simplifies further calculations when combined with other terms.

Simplification of Expressions

Simplification involves reducing an expression to its most basic form, making it easier to handle or understand. Let's look at the steps involved:

• After expanding the function using the binomial theorem, distribute constant multiples through the expanded expression. For example, we have the term with a coefficient "2" in \[2(x^3 + 9x^2 + 27x + 27)\]Multiplication distributively gives us\[2x^3 + 18x^2 + 54x + 54\]
• Next, handle the other terms, such as \(- (x + 3)\) and \(+ 4\), by distributing or directly evaluating them in context:\[-x - 3 + 4\]
• With all pieces expanded, combine like terms across the expression, where terms with similar powers of \(x\) add or subtract their coefficients.

Combining and reducing terms like \(54 - 1x = 53x\) and simplifying the constants yields the simplified function.Finally, the result of simplification gives you\[g(x) = 2x^3 + 18x^2 + 53x + 55\]This expression is much neater and is easier to use for further calculations or interpretations.