By the end of this section, you should be able to:

1. Describe the properties (domain, range, shape) of the quadratic, reciprocal, square root, and inverse functions

This section might be long, but hopefully you will find it useful. Here, we will investigate the properties of four basic functions: the quadratic, reciprocal, square root, and inverse functions. Understanding the shape and properties of these functions is at the heart of many concepts you will encounter later on.

**The Quadratic Function**: Ah yes, the smiley face function (or frowny face, depending on the transformation applied). While I'd much rather it be called a "smiley-faced function", it's actually called a**parabola**. The sketch below is that of a typical quadratic function, \(f(x) = x^2\), but the general form for the quadratic function is \(f(x) = ax^2 + bx + c\). For \(f(x) = x^2\), \(a=1, b=0, and\:c=0\). Here are a few key points regarding the quadratic function:

- The vertex (which we will denote by \((p,q)\) represents a critical number for the quadratic function. If \(a\) is negative, or \(a<0\), we have a downward parabola (a.k.a., frowny face) and its vertex is the highest point in the function. If positive, or \(a>0\), the parabola curves upwards, and the vertex is the lowest. For \(f(x) = x^2\), the vertex \((0,0)\)is the lowest point, since \(a>0\).
- The function is defined for all real x in the domain, but the range depends on the function. If \(a<0\), meaning the function is
**concave**downwards (another way to say frowny face function), then the range is all y values less than or equal to the y-value of the vertex \(\left\{y\epsilon\mathbb{R}|y\leq q\right\}\), If \(a>0\), the function is concave upwards and the range is all y values greater than or equal to the y-value of the vertex \(\left\{y\epsilon\mathbb{R}|y\geq q\right\}\). - The quadratic function can also be written as \(a(x-p)^2 + q\). If you'll recall our previous lesson on the transformations of functions, this may be somewhat familiar. In this form, the vertex \((p,q)\) is easy to identify, giving a reference point from which the function can be sketeched
- The graph is always symmetric about a vertical line through the vertex \(x = p\). Below is an illustration:

**The Square Root Function**: You might know that \(\sqrt{25} = 5 \), but what about \(\sqrt x\)? This is the square root function, \(f(x) = \sqrt x\). The most important property of the square root function is that \(x \geq0\), or "x must be greater than or equal to zero". You might recall from an earlier lesson, "types of numbers", that we represent \(\sqrt{-1}\) as \(i\), a complex number. Because we normally graph only the real numbers, complex numbers are not defined in the real number space, and are thus not graphed. Below is an example graph, specifically that of \(f(x) = \sqrt x\). For this function, the domain is \(\{ x\epsilon\mathbb{R}|x\geq0\}\) and the range is \(\{ y\epsilon\mathbb{R}|y\geq0\}\). To simplify, we can consider such a function as being of the form \(\sqrt{x-p}+q\), where the domain is \(\{ x\epsilon\mathbb{R}|x\geq p\}\)and the rnage is \(\{ y\epsilon\mathbb{R}|y\geq q\}\)

**Reciprocal Function:**In mathematics, when we ask for the reciprocal of x, we are asking for "one over" that number, or \(\frac{1}{x}\). Similar to how a square root function cannot be less than zero (i.e., negative), a reciprocal function cannot have a denominator equal to zero. This is because we cannot divide by zero. Reciprocal functions are used to denote an inversely proportional relationship. For example, for a fixed volume of gas, the number of particle collisions is inversely proportional to the size of container. In other words, particles in a smaller container experience more collisions, while those in a larger container experience fewer collisions, as shown below.

Now, away from chemistry and back to math! The reciprocal function has some features to know:

- The domain of the typical reciprocal function, \(f(x) = \frac{1}{x}\), is \(\{x\epsilon\mathbb{R}|x\neq0\}\). This is because x cannot equal zero, \(x\neq0\).
- The range of the reciprocal function \(f(x) = \frac{1}{x}\) is similar to the domain: \(\{y\epsilon\mathbb{R}|y\neq0\}\). This is because a reciprocal function with a constant numerator can never equal zero. These restrictions on the domain and range mean that the function has no x or y intercepts.
- A final interesting fact is that the reciprocal function \(f(x) = \frac{1}{x}\) is symmetric about \(y=x\) and \(y=-x\). An example function is shown below:

**The Inverse Function**: Last, but not least, comes the inverse function. In a previous lesson, we defined the function as a specific type of relation that assigns to each element in the domain a single element in the range. The inverse function can be seen as doing the reverse! That is, taking the output values and giving the original input. For example, the function \(f(x) = x^2\) is a relation that squares its input. If the input is 2, for example, the output is 4. If the input is 3, the output is 9, and so on. The inverse of this function would be one that takes an input of 9 and outputs 3, or an input of 4 and outputs 2. This type of relation is summarized in the following animation (the reverse of the animation shown in the lesson "What is a Function?")

We denote the inverse of a function \(f(x)\) as \(f^{-1}(x)\), if it is also a function. There are three main cases for which you will be asked to find the inverse:

- When the function is given as an ordered pair:

For an ordered pair \(f = \{(1,4), (2,3), (4,5), (7,7)\}\), we find the inverse by switching the places of the x and y coordinates, such that the inverse relation becomes \(f^{-1} = \{(4,1), (3,2), (5,4), (7,7) \}\). Plotting both the original and inversed points, you might observe an interesting pattern; the new points (shown in lime green) are simply the original points reflected along the line \(y=x\). This is the definition of the inverse function; a function reflected about the line \(y=x\) gives its inverse.

- When the function is given as an equation:

For the function \(f(x) = 3x - 2\):

- Replace \(y\) with \(f(x)\) and solve for \(x\):

\(y = 3x - 2\)

\(y + 2 = 3x\)

\(x = \frac{y+2}{3}\)

- Replace \(y\) and \(x\). Is the resulting relation a function? (Yes!) Replace \(y\) with \(f^{-1}\):

\(f^{-1} = \frac{x+2}{3}\)

Another, slightly tougher example:

\(4x^2 + 16y^2 = 144\)

Solving for x:

\(4x^2 = 144 - 16y^2\)

\(x^2 = \frac{144-16y^2}{4}\)

\(x = \pm\sqrt{\frac{144-16y^2}{4}}\)

\(x = \pm\sqrt{36-4y^2}\)

Replace y with x. Is this a function? No, it's only a relation as we have two \(x\)-values (as given by the \(\pm\)) for each \(y\) value. Therefore, we leave it as:

\(y = \pm\sqrt{36-x^{2}}\)

- When the relation is given without an equation.

For example, try to the inverse of the following function:

This is somewhat a trick question. As you might have recognized, this is the graph of \(f(x) = \frac{1}{x}\); the reciprocal function. You might recall from its definition that the reciprocal function is *symmetrical* about the line \(y=x\). Therefore, the inverse of the function is the function itself . If you want, you can try to prove this for yourself using the algorithm outlined in the previous example. What about other functions? Below is shown the graph of a sample function, \(f\) and its inverse, \(f^{-1}\):

A unique property of the inverse function is that its domain and range is the inverse of that of the original function. For example, for the function \(f(x) = x^2 + 2\), the inverse is:

\(y = x^2 + 2\)

\(x^2 = y- 2\)

\(y = \pm\sqrt{x-2}\) (not a function)

As we know from earlier in this lesson, because the coefficient preceding the \(x^2\) term in \(f(x) = x^2 + 2\) is \(>0\) (one, to be precise), the function is concave upwards, so its range is all points greater than or equal to the y-value of its vertex, \((0,5)\), or \(\{y\epsilon\mathbb{R}|y\geq2\}\). The domain, as with all quadratic functions, is all real numbers, \(\{x\epsilon\mathbb{R}\}\). Therefore, we expect its inverse to have the opposite domain and range. In other words, we expect the domain and range for \(y = \pm\sqrt{x-2}\) to be \(\{x\epsilon\mathbb{R}|x\geq2\}\) and \(\{y\epsilon\mathbb{R}\}\), respectively. Graphing the function and its inverse below, we see that this is true!

With that, the lesson is over. Be sure to try some of the practice flashcards. In the next lesson, we will review what we have learned in the past few lectures with a nice quiz! You guys like quizzes, right?

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