Problem 42 - Algebra 2

Question

Let \(f(x)=x^{2}\) and \(g(x)=x-3 .\) Find each value or expression. $$ (g \circ f)(-a) $$

Step-by-Step Solution

Answer

The value of \(g \circ f(-a)\) is \(a^{2} - 3\).

Step 1: Understand Function Composition

In function composition, we apply one function to the results of another function. In this case, we are asked to calculate \(g \circ f(-a)\), which means we have to apply function \(g\) to the results of function \(f(-a)\).

Step 2: Compute \(f(-a)\)

First, compute \(f(-a)\). Given that \(f(x) = x^{2}\), we substitute \(x\) with \(-a\) to get \(f(-a) = (-a)^{2} = a^{2}\).

Step 3: Substitute \(f(-a)\) into \(g(x)\)

Next, we will substitute \(f(-a)\) into \(g(x)\). Given \(g(x) = x - 3\), so we substitute \(x\) with \(a^{2}\) to get \(g(f(-a)) = a^{2} - 3\).

Key Concepts

Understanding Quadratic Functions

Quadratic functions are a cornerstone of algebra and an essential concept in mathematics. A quadratic function is one in which the highest power of the variable is two. The standard form of a quadratic function is expressed as:

• The general formula: \(f(x) = ax^2 + bx + c\)
• The leading coefficient 'a' determines the direction and "width" of the parabola's opening.
• The constant 'c' is the y-intercept, representing the function's value when \(x = 0\).

For instance, in the problem given, the function \(f(x) = x^2\) is a quadratic function:

• 'a' is 1, showing it is a basic parabola opening upwards.
• 'b' and 'c' are both zero, resulting in a vertex at the origin (0,0) with its axis of symmetry along the y-axis.

Quadratic functions often appear in equations modeling real-world phenomena such as the path of projectiles or areas compared to perimeter lengths. Understanding their algebraic form lays the groundwork for many advanced mathematical concepts.

The Process of Function Evaluation

Function evaluation is a fundamental skill in algebra where you substitute variables with actual values or expressions. In simpler terms, given a function \(f(x)\), you are tasked with finding the output (or result) for a specific input.
For example, if you have \(f(x) = x^2\), evaluating the function for \(x = -a\) means replacing \(x\) with \(-a\). So, \(f(-a) = (-a)^2\), which simplifies to \(a^2\).

• Substitution replaces the variable with given values or expressions.
• Careful computation, especially with negative values, ensures correct outcomes.
• Working step-by-step prevents errors and reinforces understanding.

Evaluating functions accurately is a crucial part in problems involving function composition and many other areas of mathematics, like derivatives in calculus.
It provides the functional outputs that are necessary for further computations.

Exploring Function Composition

Function composition occurs when you apply one function on the results of another. It is represented as \((g \circ f)(x)\), which translates to applying \(f(x)\) and then passing its output into \(g(x)\). This concept allows us to combine functions to create new ones.

• Order matters; \((g \circ f)(x)\) is not the same as \((f \circ g)(x)\).
• Start with the innermost function and proceed outward, ensuring each step's result is correctly computed.

In our exercise, we start with calculating \(f(-a)\):

• From \(f(x) = x^2\), substituting gives \(f(-a) = a^2\).
• This result, \(a^2\), is then input into \(g(x) = x - 3\).
• Finally, substituting \(a^2\) into \(g\) yields \(g(a^2) = a^2 - 3\).

Function composition helps form complex mathematical models and facilitates transitions between variable inputs or changes in functions.