Problem 61 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression for \(f(x + h)\) is \(2x^2 - 5x + 4xh + 2h^2 - 5h + 1\).

Step 1: Understand the Task

We're asked to find and simplify the expression \(f(x + h)\) for the function \(f(x) = 2x^2 - 5x + 1\). This means we need to substitute \(x + h\) into the function wherever there is an \(x\).

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

Start by substituting \(x + h\) into the function \(f(x)\):\[f(x + h) = 2(x + h)^2 - 5(x + h) + 1\]Next, we need to expand \((x + h)^2\) and \(-(5)(x + h)\).

Step 3: Expand the Squared Term

Expand \((x + h)^2\):\[(x + h)^2 = x^2 + 2xh + h^2\]Substitute this back into the equation:\[f(x + h) = 2(x^2 + 2xh + h^2) - 5(x + h) + 1\]

Step 4: Distribute and Simplify the Expression

Distribute the 2 into the expanded squared term and the -5 into \((x + h)\):\[= 2x^2 + 4xh + 2h^2 - 5x - 5h + 1\]Now, combine like terms.

Step 5: Combine Like Terms

Group and combine the like terms:\[f(x + h) = 2x^2 - 5x + 4xh + 2h^2 - 5h + 1\]There are no further like terms to combine, so this is our simplified expression for \(f(x + h)\).

Key Concepts

Polynomial Functions

Polynomial functions are a type of algebraic expression. They are composed of variables and coefficients that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the example given, the function is a polynomial of degree 2, expressed as:

• \( f(x) = 2x^2 - 5x + 1 \)

The highest power of the variable \( x \) is 2, indicating that this is a quadratic function, which is a specific type of polynomial function. Quadratic polynomials have a variety of applications in real-world scenarios, such as calculating areas, determining profits, and modeling the paths of objects in physics.

Function Evaluation

Function evaluation involves substituting a specific value into a function in place of its variable. It often helps us understand how the function behaves for different values. The problem requires evaluating the function at a modified input \( x + h \). This means replacing \( x \) everywhere in the original function \( f(x) = 2x^2 - 5x + 1 \) with \( x + h \). By doing this, we create:

• \( f(x + h) = 2(x + h)^2 - 5(x + h) + 1 \)

Function evaluation is crucial for analyzing polynomial functions in a variety of contexts, such as in business modeling or scientific calculations.

Simplification

Simplification refers to the process of making an expression easier to understand or work with. In algebra, it means to perform all possible operations to reduce an expression to its simplest form. For the expression

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

this involves distributing and combining like terms.

• First, expand the squared term \((x + h)^2 = x^2 + 2xh + h^2 \).
• Then distribute: \(2(x^2 + 2xh + h^2)\) becomes \(2x^2 + 4xh + 2h^2\).
• Negative distribution: \(-5(x + h) = -5x - 5h \).
• Finally, combine all like terms to achieve the simplified expression: \( 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 \).

Expansion of Expressions

An essential technique in algebra is the expansion of expressions, which involves multiplying out brackets to simplify the expression. In the task, we expanded the expression

• \((x + h)^2 = x^2 + 2xh + h^2\)

This step was crucial for substituting into the function \( f(x + h) \) and demonstrated how we can turn a product into a sum of terms. Expansion requires understanding of distributive properties and allows for further simplification of algebraic expressions. By expanding an expression, we reveal the individual components or terms, making it easier to combine and simplify them into a more manageable form. It's a fundamental skill when dealing with polynomial functions or any complex algebraic operations.