FUNCTION (IGCSE/GCE O Level ADD MATHS) 0606

Summary

In this video, we delve into key concepts of functions, starting with understanding their definition, domain, and range. The lesson explores how to identify functions from graphs, explain the domain and range, and how restrictions affect these properties. We also cover inverse functions, emphasizing when they exist and how to find their domain and range. Lastly, composite functions are introduced, with practical examples of how to determine whether they exist based on the inner and outer functions. Throughout, various types of problems are tackled, from simple to complex, ensuring a comprehensive grasp of functions in mathematics.

Takeaways

• 😀 Functions relate X and Y values, where each X value corresponds to only one Y value to be considered a function.
• 😀 To identify a function on a graph, use the vertical line test: if a vertical line intersects the graph at more than one point, it's not a function.
• 😀 The domain of a function refers to the allowed X values (left-right on the graph), while the range refers to the allowed Y values (up-down on the graph).
• 😀 A function with no restrictions on left or right is said to have a domain of all real numbers (X ∈ ℝ).
• 😀 The range of a function describes the vertical values, and can have a minimum or maximum depending on the graph’s behavior.
• 😀 If a function's domain is restricted, the range is also affected accordingly.
• 😀 To determine if a quadratic function has an inverse, restrict the domain to make it a one-to-one function.
• 😀 A function has an inverse only if it is one-to-one, meaning each X value has one Y value and vice versa.
• 😀 The domain and range for the inverse function are swapped compared to the original function.
• 😀 When drawing the inverse function on the same graph, it is reflected over the line y = x in relation to the original function.
• 😀 Composite functions involve combining two functions, and the domain of the composite function depends on the inner function’s domain and the outer function’s range.

Q & A

What is the definition of a function?

A function relates each x-value to exactly one y-value. If an x-value corresponds to more than one y-value, it is not a function.

How can you determine if a graph represents a function?

You can use the vertical line test. If a vertical line intersects the graph at more than one point, it is not a function.

What is the difference between domain and range?

The domain refers to the set of all possible x-values of a function, representing its left-right movement. The range refers to the set of all possible y-values, representing the graph's up-down movement.

What does it mean if a graph's domain and range are unrestricted?

If a graph's domain and range are unrestricted, it means the function can continue to extend indefinitely to the left and right (domain), and upward and downward (range) without any limitations.

How do you find the domain and range for a function with a restricted domain?

For a restricted domain, you first identify the specific x-values given in the problem. The range is then determined based on how the function behaves with that restricted domain.

What is the condition for a function to have an inverse?

A function must be one-to-one, meaning that each x-value corresponds to exactly one unique y-value and vice versa. This is necessary for a function to have an inverse.

How do you find the inverse of a function?

To find the inverse, swap the x and y values in the original equation and solve for y to express the inverse function.

What is the relationship between the graph of a function and its inverse?

The graph of a function and its inverse are symmetrical with respect to the line y = x.

How do you handle composite functions?

To handle composite functions, substitute the output of one function into the input of another. The domain of the composite function is determined by the domain of the inner function.

What is an example of a reciprocal function and its behavior?

A reciprocal function, such as 1/x, may have an asymptote. For example, if x = 2.5, the graph may not intersect at that point due to the asymptote.