Problem 75 - College Algebra with Modeling and Visualization

Question

If the points \((-5,-6)\) and \((-3,4)\) lie on the graph of an odd function \(f,\) then what \(\operatorname{do} f(5)\) and \(f(3)\) equal?

Step-by-Step Solution

Answer

\(f(5) = 6\) and \(f(3) = -4\).

Step 1: Understand the Properties of Odd Functions

An odd function, by definition, satisfies the condition \(f(-x) = -f(x)\) for all \(x\). This means if a point \((x, y)\) is on the graph, then \((-x, -y)\) is also on the graph.

Step 2: Apply Odd Function Property to Given Points

We have the points \((-5, -6)\) and \((-3, 4)\). Since the function is odd, \(f(5) = -f(-5)\) and \(f(3) = -f(-3)\).

Step 3: Calculate \(f(5)\)

From the point \((-5, -6)\), using the property of odd functions, \(f(5) = -(-6) = 6\).

Step 4: Calculate \(f(3)\)

From the point \((-3, 4)\), using the property of odd functions, \(f(3) = -(4) = -4\).

Key Concepts

Algebra

Algebra serves as the foundation for understanding odd functions. Odd functions have a special algebraic property: for any given function \(f\), the rule \(f(-x) = -f(x)\) defines it as odd. This means that if you substitute \(-x\) into the function, you should get the negative of what you get when you substitute \(x\). This property creates a symmetry around the origin in the coordinate plane.

When dealing with algebraic expressions of functions, recognizing this symmetry can simplify calculations. For example, if given a point \((-5, -6)\) on the graph of an odd function, you can immediately determine the function value at the opposite point, \(f(5)\). With our understanding of this relationship, \(f(5) = 6\) because it's the opposite of \(f(-5)\). This symmetry and algebraic manipulation help solve problems efficiently in algebra.

Function Properties

Function properties, such as the oddness of a function, play a crucial role in determining the values of functions across different points. The defining characteristic of an odd function is its symmetrical property around the origin, which means the function is reflective over both axes when the graph is drawn.

This property helps when you're given specific coordinates, such as \((-3, 4)\) and need to find \(f(3)\). Given the property \(f(-x) = -f(x)\), the value becomes immediately clear: \(f(3) = -f(-3)\), thus \(f(3) = -4\).

Understanding these properties is not just about knowing rules, but using them effectively in problems, which streamlines the process of solving functional equations.

Coordinate Geometry

Coordinate geometry allows us to visually interpret the properties of odd functions. When plotted, the graph of an odd function displays rotational symmetry about the origin. This symmetry is a direct consequence of the algebraic condition \(f(-x) = -f(x)\).

Consider the points \((-5, -6)\) and \((-3, 4)\) on the graph. Using coordinate geometry, you can predict their symmetrical counterparts: \((5, 6)\) and \((3, -4)\). This visual symmetry helps confirm the calculations you perform algebraically. The graph shows that if you rotate a point \((x, y)\) 180 degrees about the origin, it lands on the point \((-x, -y)\).

Understanding the visualizations in coordinate geometry not only aids in grasping algebraic properties but also makes solving problems involving function properties more intuitive. It provides a deeper understanding of how mathematical principles unfold on a plane.