So, in grade 12 they thought long and hard about functions until they realised they couldn’t add anything to grade 11 functions. I don’t know who, but someone said let’s reflect them around the mirror line y=x and they did that, just that.

An inverse function is the reflection of that function through the line y=x, every x coordinated is swapped with its corresponding y coordinate. For example, the inverse of (2;7) will be (7;2) and if we connect all the swapped coordinates we end up with an inverse function. All the functions can have possible inverse, except for the parabola. We have to restrict the domain of the parabola in order to have its function, owing to the vertical line test.

To solve for the inverse equation, we simply swap x and y in the given equation and the solve for y, or make y subject of the formula. Learn the notation of the inverse formula from your study guide. In the case of the exponential graph we will have to solve the exponent using logarithm (if y=b^x, then x=log(b)(y)). We can use this formula to quickly plot the graph using the following methods:

• Two points method for the straighline graph
• Turning point and intercepts for the parabola
• Asymptote and intercept or other point for the exponential graph, the x-asymptote is swapped to become a y-asymptote.
• Well, what do you think the inverse of the hyperbola is? It already has its reflection because the function on the second quadrant is reflected through y=x to make the function on fourth quadrant. So, we won’t bother about the hyperbola, lets leave it as it is.

Vertical line test:

• This line is drawn like every vertical line, but dotted.
• If it touches the graph once, then that graph is a function.
• But if it touches the graph twice, we don’t have a valid function. This is the case with the parabola and we have to restrict one symmetric half from the axis of symmetry, or from the turning point, either to the right or left. After this domain restriction we will be left with a valid function.

Horizontal line test:

• The horizontal line test is used to identify the type of a function, before we talk further about this one, lets take care of the types of functions:
• Types of functions:

One-to-one function: one x value substituted to the function’s formula will result in one y value

Note: one x value will never have two possible y values, this will fail when we test it using vertical line test

Many-to-one function: two or more x values share one y value.

• The horizontal line will result in a one-to-one function if it touches the graph once.
• And obvious a many-to-one function if it touches the graph twice (parabola case) or more (cubic function)

There we go, that’s it for the inverse functions. Let’s do examples and questions on videos to follow, find the next post dealing with financial mathematics.

This blog post is not the first step towards learning the concepts of math but the coupling step to ease the use of your study guide and other study material. Read it over and over again to keep the concepts at the back of your head. Math is a subject of rules; know the rules, be able to duplicate them on a blank paper and go fetch your distinction