Introduction to Functions (2 of 2: Examples & Counter-Examples)

Summary

In this video, the speaker explores the concept of functions and relations in mathematics. They introduce the idea of inputs (X) and outputs (Y), explaining how functions relate these with a one-to-one correspondence, meaning each input has exactly one output. The video contrasts this with relations, which may have multiple outputs for a single input, like the equation of a circle. To determine if a graph represents a function, the vertical line test is introduced, where a graph must not be intersected by a vertical line at more than one point. This engaging lesson breaks down key concepts for students to understand the difference between functions and relations.

Takeaways

• 😀 Inputs and outputs are fundamental concepts in mathematics, typically represented by X (input) and Y (output).
• 😀 The notation 'y as a function of x' represents a relationship where the value of Y depends on X.
• 😀 A function is a special type of relation where each input (X) has exactly one output (Y).
• 😀 For the equation y = (x + 1)(x - 2), the output (Y) changes depending on the input (X).
• 😀 A function can be visualized as a curve on a graph, with X values determining the Y values at each point.
• 😀 If an equation results in more than one output for a single input, it is not a function, even though a relationship exists.
• 😀 An example of a non-function is the equation of a circle, where some X values correspond to two Y values.
• 😀 The vertical line test is a method used to determine if a graph represents a function. If a vertical line crosses the graph more than once, it’s not a function.
• 😀 Functions must have exactly one Y for each X in their domain, and this uniqueness is essential for the definition of a function.
• 😀 The circle equation (x^2 + y^2 = 1) demonstrates a case where the same X value (like 0) can produce two different Y values (positive and negative), hence it’s not a function.

Q & A

What are the classic labels given to inputs and outputs in mathematical functions?

The classic labels for inputs and outputs in mathematical functions are X and Y, respectively. X is typically the input, and Y is the output.

How is the notation 'y as a function of x' represented?

'y as a function of x' is represented by the notation 'y = f(x)', where Y is the output that depends on the input X.

Why is the equation of the parabola x^2 + 1x - 2 used as an example?

The equation x^2 + 1x - 2 is used as an example because it is a simple quadratic function that illustrates the concept of roots and the relation between inputs (X) and outputs (Y).

How can you determine if a relation is a function?

A relation is a function if each input (X) has exactly one output (Y). This means for any given value of X, there must be only one corresponding Y.

What does it mean for an input to have more than one output, as seen in the circle example?

When an input has more than one output, it breaks the rule of a function, which requires that each input be associated with exactly one output. This can be seen in the example of the circle, where an input (like X = 0) gives two outputs (Y = 1 and Y = -1).

What is the vertical line test and how does it determine if a graph represents a function?

The vertical line test is a method to check if a graph represents a function. If a vertical line intersects the graph at more than one point, the graph does not represent a function. If the vertical line intersects at only one point, the graph is a function.

Why is the circle equation x^2 + y^2 = 1 not considered a function?

The equation x^2 + y^2 = 1 is not a function because, for some values of X (like X = 0), there are two corresponding values of Y (Y = 1 and Y = -1), violating the rule that a function must have exactly one output for each input.

In the context of the given transcript, how is the input 'x = -1' used to determine the output?

When the input 'x = -1' is used in the equation, the value of Y becomes 0, as the expression x + 1 becomes zero. This demonstrates that X = -1 leads to a single output, Y = 0, in a function.

How does the concept of roots relate to the function example provided?

The concept of roots is demonstrated by finding the values of X where the function equals zero. In the example, the roots of the equation x^2 + 1x - 2 are at X = -1 and X = 2, which are the points where the function crosses the X-axis.

Can a graph represent a function if some points have two outputs for a single input?

No, a graph cannot represent a function if some points have more than one output for a single input. For instance, the circle equation fails the vertical line test because some inputs, like X = 0, yield multiple outputs (Y = 1 and Y = -1), which makes it not a function.