Problem 54 - Algebra A Combined Function

Question

If \(f(x)=x^{2}-3 x,\) find each function value. $$ f(a+h) $$

Step-by-Step Solution

Answer

\(f(a+h) = a^2 + 2ah + h^2 - 3a - 3h\).

Step 1: Understand the function

The function given is \(f(x) = x^2 - 3x\). We need to find \(f(a+h)\), which means replacing \(x\) with \((a+h)\) in the function formula.

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

Replace every instance of \(x\) in the function \(f(x) = x^2 - 3x\) with \(a+h\). This gives us: \(f(a+h) = (a+h)^2 - 3(a+h)\).

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

Using the formula for expanding a square, \((a+h)^2 = a^2 + 2ah + h^2\). Substitute this into the function equation: \(f(a+h) = a^2 + 2ah + h^2 - 3(a+h)\).

Step 4: Distribute the \(-3\) across \((a+h)\)

Distribute \(-3\) through the expression \((a+h)\), giving us \(-3a - 3h\). The expression is now: \(f(a+h) = a^2 + 2ah + h^2 - 3a - 3h\).

Step 5: Simplify the expression

Combine like terms, though in this case, no terms combine: \(f(a+h) = a^2 + 2ah + h^2 - 3a - 3h\). This is the simplified expression for \(f(a+h)\).

Key Concepts

Understanding Polynomial Functions

A polynomial function is an expression involving a variable and coefficients that are combined using addition, subtraction, multiplication, and non-negative integer exponents. In our original exercise, the function given is \(f(x) = x^2 - 3x\). This is a polynomial of degree 2, often called a quadratic polynomial, because the highest exponent is 2. Here's why polynomial functions are important:

• They can model various physical situations, like projectile motion or growth trends.
• Polynomial functions are smooth and continuous, meaning they don’t have sharp turns or breaks.
• They are simple to differentiate, making them useful in calculus for finding maxima, minima, and points of inflection.

Understanding polynomial functions helps in grasping more complex functions and provides a foundation for algebraic problem solving.

Breaking Down Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and subtraction). Each part of the expression separated by a plus or minus sign is called a "term." For instance, in \((a+h)^2 - 3(a+h)\), the terms are \(a^2\), \(2ah\), \(h^2\), \(-3a\), and \(-3h\). These expressions are fundamental for creating and solving equations, and they are used in various fields such as physics and engineering to describe real-world scenarios. Here is what simplifies working with algebraic expressions:

• Organizing terms helps us recognize patterns and relationships among variables.
• Distributive properties allow us to expand and rearrange expressions for clarity and simplicity.
• The rules of arithmetic operations help bring the expressions to their simplest form.

As we manipulate these expressions, we practice critical algebraic skills that are crucial for solving equations and inequalities.

Exploring Function Substitution

Function substitution involves replacing a variable in a function with a new expression. It's like substituting ingredients in a recipe to see how they change the outcome. In the step-by-step solution, we performed function substitution by evaluating \(f(a + h)\) from \(f(x) = x^2 - 3x\).Function substitution is a powerful tool in mathematics because:

• It allows us to explore how changes in variables affect the entire function.
• It helps in computing derivatives in calculus where substituting expressions is common.
• It is used in real-world applications, such as calculating compound interest or analyzing population growth trends.

By understanding function substitution, students develop deeper insights into how functions behave and how they can be manipulated, paving the way for more advanced mathematical concepts.