Problem 61 - Brief Applied Calculus

Question

$$ \text { For each function, find and simplify } f(x+h) \text { . } $$ $$ f(x)=2 x^{2}-5 x+1 $$

Step-by-Step Solution

Answer

\( f(x+h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \).

Step 1: Understand the Problem

We are given a function \( f(x) = 2x^2 - 5x + 1 \). The task is to find \( f(x+h) \) where \( h \) is another variable, and then simplify the expression.

Step 2: Substitute \( x+h \) into the function

Replace every instance of \( x \) in the function \( f(x) \) with \( x + h \). This yields:\[ f(x+h) = 2(x+h)^2 - 5(x+h) + 1 \].

Step 3: Expand \( (x+h)^2 \)

Compute \( (x+h)^2 \) using the binomial expansion:\[ (x+h)^2 = x^2 + 2xh + h^2 \].Substitute this back into the function:\[ f(x+h) = 2(x^2 + 2xh + h^2) - 5(x+h) + 1 \].

Step 4: Distribute and Simplify Terms

Distribute the 2 into the expanded quadratic and simplify:\[ f(x+h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \].Combine any like terms if possible: there are none to combine here.

Step 5: Final Function Check

Double-check your expansion and distribution for accuracy. Verify that the expression corresponds to the steps taken and is fully simplified.

Key Concepts

Function Substitution

Function substitution is a key concept in understanding how functions work, especially when variables are introduced or adjusted. In this exercise, we're working with a function, \( f(x) = 2x^2 - 5x + 1 \). Our task is to replace every instance of \( x \) in the function with \( x+h \).

This substitution transforms the function into \( f(x+h) \). It is like finding the function's value at a point shifted by \( h \). It helps us determine how the output of the function changes as the input changes by \( h \).

• Replace each \( x \) with \( x+h \) in the function's expression.
• In our example: \( f(x+h) = 2(x+h)^2 - 5(x+h) + 1 \).

This new expression is not simplified yet, but the substitution gives us a starting point.

Polynomial Expansion

Once the substitution is done, we move to polynomial expansion, an essential step when dealing with expressions like \( (x+h)^2 \). Here, we expand \( (x+h)^2 \) using the binomial theorem.

The binomial theorem tells us how to expand powers of sums. So, \((x+h)^2 = x^2 + 2xh + h^2 \). The expansion explains how each term in \( x+h \) interacts with itself when squared.

• Start with \((x+h)^2\), which means multiplying \( (x+h) \, \times \, (x+h) \).
• Distribute each term: \( x \times x = x^2 \), \( x \times h = xh \), \( h \times x = hx \), \( h \times h = h^2 \).

Adding these results gives the expanded form \(x^2 + 2xh + h^2 \). After expansion, simplify further by distributing any factors outside the parentheses into the newly expanded terms.

Algebraic Simplification

Algebraic simplification involves combining like terms and making an expression as simple as possible. After expanding, we have the expression: \[ f(x+h) = 2(x^2 + 2xh + h^2) - 5(x+h) + 1 \].

Let's distribute the constants through the expression to simplify it step by step:

• First, multiply \( 2 \) by each term inside \( (x^2 + 2xh + h^2) \): \( 2x^2 + 4xh + 2h^2 \).
• Next, distribute \( -5 \) through \( (x+h) \): \(-5x - 5h \).

Combine these results with the constant \( 1 \) to obtain:
\[ f(x+h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \].

No like terms remain, so this is the simplified form of \( f(x+h) \). Ensuring each part of the expression is accurately expanded and simplified is critical for correct results. This process highlights the importance of careful arithmetic and consistent checking.