Function Operations and Properties

The video tutorial covers combining functions, shifting, and scaling graphs. It begins with algebraic manipulations, explaining how adding, subtracting, multiplying, and dividing functions affect their domains. The tutorial then moves to composite functions, illustrating how to determine their domains. Next, it discusses shifting graphs vertically and horizontally, followed by scaling and reflecting graphs. Examples are provided throughout to clarify these concepts.

What is the primary focus of this video tutorial?

• Understanding the history of mathematics

• Learning about combining, shifting, and scaling functions

• Studying the properties of geometric shapes

• Exploring the applications of calculus

What happens when you add two functions together?

• You get a new function with a domain that is the union of the original domains

• You get a new function with a domain that is the intersection of the original domains

• The original functions remain unchanged

• The new function has no domain

When subtracting two functions, what is the domain of the resulting function?

• The domain of the first function only

• The domain of the second function only

• The union of the domains of the original functions

• The intersection of the domains of the original functions

What must be true for the division of two functions to be valid?

• Both functions must be linear

• The numerator must be greater than the denominator

• The functions must have the same domain

• The denominator function must not be zero

What is a composite function?

• A function that is the difference of two functions

• A function that is the sum of two functions

• A function that is the product of two functions

• A function that is formed by applying one function to the result of another function

In a composite function f(g(x)), what must be true about the domain?

• The domain is the union of the domains of f and g

• x must be in the domain of f

• g(x) must be in the domain of f

• x must be in the domain of both f and g

What does a vertical shift of a graph involve?

• Moving the graph left or right

• Reflecting the graph across the x-axis

• Moving the graph up or down

• Stretching the graph horizontally

How does a positive horizontal shift affect a graph?

• It stretches the graph vertically

• It shifts the graph to the right

• It shifts the graph to the left

• It compresses the graph horizontally

What effect does multiplying a function by a constant greater than one have?

• It stretches the graph vertically

• It reflects the graph across the y-axis

• It shifts the graph horizontally

• It compresses the graph vertically

What happens when a function is divided by a constant?

• The graph is stretched vertically

• The graph is compressed vertically

• The graph is reflected across the x-axis

• The graph is shifted upwards