Problem 73 - A Graphical Approach to Precalculus with Limits

Question

For each function find (a) \(f(x+h)\) and (b) \(f(x)+f(h)\) $$f(x)=3 x-x^{2}$$

Step-by-Step Solution

Answer

(a) \(f(x+h) = 3x + 3h - x^2 - 2xh - h^2\); (b) \(f(x) + f(h) = 3x + 3h - x^2 - h^2\).

Step 1: Substitute in the expression for \(f(x+h)\)

To find \(f(x+h)\), substitute \(x+h\) into the function \(f(x) = 3x - x^2\). This gives us:\[f(x+h) = 3(x+h) - (x+h)^2\].

Step 2: Expand and simplify \(f(x+h)\)

Now, expand the expression:- Expand \(3(x+h)\) to get \(3x + 3h\).- Expand \((x+h)^2\) using \((x+h)(x+h) = x^2 + 2xh + h^2\).Substituting back into the equation: \[f(x+h) = 3x + 3h - (x^2 + 2xh + h^2)\]Simplify:\[f(x+h) = 3x + 3h - x^2 - 2xh - h^2\].

Step 3: Find \(f(x) + f(h)\)

Calculate \(f(h)\) using the given function by substituting \(h\) for \(x\):\[f(h) = 3h - h^2\].Now, find \(f(x) + f(h)\) by adding the expressions for \(f(x)\) and \(f(h)\):\[f(x) + f(h) = (3x - x^2) + (3h - h^2)\].

Step 4: Simplify \(f(x) + f(h)\)

Combine the terms in the expression calculated previously:- Combine the \(3x\) and \(3h\) terms: \(3x + 3h\).- Combine the negative squared terms: \(- x^2 - h^2\).This results in:\[f(x) + f(h) = 3x + 3h - x^2 - h^2\].

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables multiplied by coefficients. A polynomial in one variable, like the given function \(f(x) = 3x - x^2\), consists of terms where the variable \(x\) is raised to a non-negative integer power. The function given is a quadratic polynomial because its highest degree is 2, corresponding to the term \(-x^2\). Each polynomial term can typically have:

• A coefficient, which is a number multiplying the variable (- In \(3x\), the coefficient is 3.
  - In \(-x^2\), the coefficient is -1.)
• A degree, which is the power to which the variable is raised (- \(3x\) is a degree of 1.
  - \(-x^2\) is a degree of 2.)

To solve problems involving polynomials, you must be able to substitute variables and manipulate the algebraic expressions to expand and simplify the results.

Function Operations

Function operations involve different ways to manipulate functions such as evaluating, combining, adding, subtracting, or even dividing them. In this exercise, we focus on evaluating a function with different inputs and then combining their results. Understanding function operations helps us:

• Find new functions by manipulating existing ones (- Here, we needed to find \(f(x+h)\) by substituting \(x+h\) into the original function.)
• Evaluate and simplify results (- After substitution, expanding the polynomial terms like \(3(x+h)\) and \((x+h)^2\) allows calculation of \(f(x+h)\).)
• Add function values (- Finding \(f(x) + f(h)\) required calculating both \(f(x)\) and \(f(h)\) independently and then combining them.)

Practicing these operations strengthens your ability to handle myriad mathematical scenarios involving functions.

Algebraic Expressions

Algebraic expressions form the base of understanding functions and how they behave. They consist of variables, numbers, and operations like addition or multiplication, and require simplification and manipulation. With respect to this exercise, managing algebraic expressions involved:

• Substitution: Replacing the variable \(x\) with \(x+h\) transforms the expression and provides new insights into how a function behaves when shifted.
• Expansion: Distributing terms in expressions like \(3(x+h)\) ensures all aspects of the function's change are captured.
• Simplification: Combining like terms allows for a cleaner expression, such as bringing together \(3x + 3h\) and \(-x^2 - h^2\).

Mastering algebraic expressions is crucial as they form the basis for modeling real-world problems with functions. This skill lets you break down complex mathematical scenarios into manageable parts for easier understanding and solution.