Problem 77 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\(f(x+h) = 3x - x^2 + 3h - 2xh - h^2\) and \(f(x) + f(h) = 3x + 3h - x^2 - h^2\).

Step 1: Evaluate f(x+h)

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

Step 2: Expand f(x+h)

Now, expand the expression obtained in the previous step: \[f(x+h) = 3x + 3h - ((x)^2 + 2xh + (h)^2)\] \[f(x+h) = 3x + 3h - x^2 - 2xh - h^2\].

Step 3: Simplify f(x+h)

Combine like terms to simplify \(f(x+h)\): \[f(x+h) = 3x - x^2 + 3h - 2xh - h^2\].

Step 4: Calculate f(x) + f(h)

Now, find \(f(x) + f(h)\) by calculating each separately. Start with \(f(h)\), which is evaluated as: \[f(h) = 3h - h^2\]. So, \(f(x) + f(h)\) becomes: \[f(x) + f(h) = (3x-x^2) + (3h-h^2)\].

Step 5: Simplify f(x) + f(h)

Combine the expressions: \[f(x) + f(h) = 3x + 3h - x^2 - h^2\].

Key Concepts

Algebraic Expressions

An algebraic expression is essentially a mathematical phrase that contains numbers, variables, and operations. It doesn't need an equal sign like an equation does. Consider it a way to describe real-world scenarios mathematically. For instance, in the exercise's function, we have the expression \(3x - x^2\), which represents the relationship between input \(x\) and output, based on the operations of multiplication, subtraction, and exponentiation.

The beauty of algebraic expressions is in their flexibility. They can be manipulated by performing operations such as addition, subtraction, multiplication, and division. This ability to transform and simplify is what allows us to solve problems and uncover insights. In practical terms, simplifying an expression like \(3x - x^2\) helps us understand its behavior, particularly when dealing with variables and their potential values. Developing skill with algebraic expressions is foundational for tackling more complex mathematical challenges.

Polynomials

Polynomials are a special type of algebraic expression that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In our exercise, the function \(f(x) = 3x - x^2\) is a polynomial. Specifically, it's a second-degree polynomial, also known as a quadratic polynomial, because the highest power of the variable \(x\) is two.

Polynomials are fundamental because they can represent a wide range of real-world situations and can be graphed as smooth curves. Each polynomial has a degree, determined by the highest exponent of its variable(s). Understanding polynomials is important because they form the foundation for more advanced topics like calculus and are essential in various applications from engineering to economics.

They also possess properties such as roots, which are solutions to the equation formed when the polynomial is set equal to zero, and can be easily added, subtracted, and multiplied, which is useful in solving larger systems of equations.

Function Operations

Function operations involve performing mathematical tasks on functions, similar to operations on numbers and expressions. In this context, we look at evaluating functions, as seen in the exercise where we evaluate \(f(x+h)\) and \(f(x) + f(h)\). To evaluate \(f(x+h)\), we substitute \(x+h\) in place of \(x\) in the given function, leading us to perform both substitution and simplification operations.

Function addition involves finding sums of function outputs as shown by evaluating \(f(x) + f(h)\). It's about handling each part of the function separately and then combining results. This reveals relationships between varying and fixed values, showing how changes in input affect corresponding changes in output.

• **Substitution:** Insertion of new values into a function like substituting \(x + h\) for \(x\);
• **Simplification:** Combining like terms to streamline expressions, making them easier to work with;
• **Addition/Subtraction:** Adding or subtracting whole functions or parts of them as needed.

Understanding and mastering these operations opens doors to exploring complex systems and equations, providing insights that facilitate further mathematical exploration.