#Introduction to functions

The first two terms you can come across regarding functions are the domain and range. This onepager tells you what they are and how you can determine them.

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#Introduction to functions

State the largest possible domain of:

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**Function as a computer**

$f\left(x\right)$ is like a computer, you give in an $x$-value, it computes and gives back a $y$-value. The set of values that can be given in, the possible $x$-values are the domain. Similarly the set of values that come out of a function, the possible $y$-values are called the range.

**How to find the domain?**

We are working on the set of real numbers, this is our base set. For the correct solution we need to check some rules and take out the incorrect numbers from the base set.

**Division by $0$ **

If we see a fraction, the first thing we need to do is to check the denominator whether it contains an $x$ (unknown) or not. If it does contain the $x$, let's equate the denominator with $0$ and solve for $x$. The solution is a value that cannot be substituted into the function, therefore the domain is going to be all real numbers, except this specific value.

**What is the largest possible domain of $f\left(x\right)=\frac{1}{x}$, $x\in \mathrm{\mathbb{R}}$?**

$f\left(x\right)=\frac{1}{x}$, this is a simple fraction.

**1st step**: the rule which we have learned above said that we cannot divide by $0$ so we have to take it out. In other words $x$, the denominator couldn’t be equal with $0$.

**2nd step**: $x\ne 0$.

The domain is all of the numbers except $0$.

**What is the largest possible domain of $g\left(x\right)=\frac{1}{2x+3},x\in \mathrm{\mathbb{R}}$?**

$g\left(x\right)=\frac{1}{2x+3},$ now we have a bit more complicated fraction and this is how you can find the domain of it:

**1st step**: you have to find the keypoint and identify the rule of the keypoint and apply: in this case the problem is that we can not divide by $0$ so the denominator of the fraction should not be equal with $0:2x+3\ne 0$

**2nd step**: solve the equation to $x$ :

subtract $+3$ from both sides:

$2x\ne -3$

divide by $2$ :

$x\ne \frac{-3}{2}$

And we found the domain of our function, it contains all real numbers except $-\frac{3}{2}$.

**Negative under the square root**

You can only take the square root of positive numbers and $0$. This is why we need to check what's under the square root. If it contains an $x$ let’s make an inequality, so that whatever is under the square root is greater than or equal to $0$. The solution of the inequality will be the domain of the function.

**What is the largest possible domain of $f\left(x\right)=\sqrt{x},x\in \mathrm{\mathbb{R}}$?**

**1st step**: as we know we cannot take the square root of negative numbers

**2nd step**: $x\ge 0$

Our domain contains all of the positive numbers and zero.

**What is the largest possible domain of $g\left(x\right)=\sqrt{4x-9}$, $x\in \mathrm{\mathbb{R}}$?**

**1st step**: find the keypoint: it is the square root, identify the rule of the square root and apply: makes the expression under the square root garter or equal to zero:

$4x-9\ge 0$

**2nd step**: solve the inequality to $x$:

add $+9$ to both of sides of the equation

$4x\ge 9$

divide by $+4$ both of the sides

$x\ge \frac{9}{4}$

and now our domain contains numbers which are greater than $\frac{9}{4}$

** Negative numbers or $0$ in the logarithm**

This comes from the definition of the logarithm. If we see an $x$ inside of the logarithm we need to make the expression strictly greater than $0$ and solve it. The solution of the inequality will define to us the domain of the function.

**What is the largest possible domain of $f\left(x\right)=\mathrm{log}\left(x\right)$ ?**

**1st step**: we can not take the logarithm of a negative number so let’s check the inside of the logarithm.

**2nd step**: $x>0$

Our domain contains all of the numbers which are strictly greater than $0$.

**What is the largest possible domain of $g\left(x\right)=\mathrm{log}(12x-3x+4)$, $x\in \mathrm{\mathbb{R}}$?**

**1st step**: find the key point and identify the rule of it and apply: in this case we have a logarithm and this function have to be greater than $0$:

$12x-3x+4>0$

**2nd step**: solve it to $x$:

$9x+4>0$

add $-4$ to both of the sides

$9x>-4$

divide by $9$

$x>-\frac{4}{9}$

and now we found the the set of domain: all of the numbers which are greater than $-\frac{4}{9}$ .

**How to find the range?**

When looking for the range, you have to figure out what values $f\left(x\right)$ can possibly take. In order to do that there are $3$ different approaches, but actually all of these just help you to get a general idea of what the function looks like and then determine what the range is.

**Sketch $f\left(x\right)$**

In case of a well-known function you already have a general idea of what the function must look like.

For example if there's a quadratic function, you know that you are working with a parabola. In order to find the range, you have to use the function's known properties and shape and find any specific values if needed.

For the quadratic function example, you have to decide whether the parabola is facing upwards or downwards and then find the max/min point. From here you will be able to write down the set of $y$-values that the function takes.

**Find the range of $f\left(x\right)={x}^{2}+3$, if the domain is $x\in \mathrm{\mathbb{R}}$ !**

**1st step**: find the keypoint: this is a quadratic function, which means it has a max or min point. The coefficients of the ${x}^{2}$ is positive ($+1$) so we will have a maximum point.

**2nd step**: we are looking for the $y$-values, substitute for example $0$.

$y={0}^{2}+3=3$

if the $x$-value is $0$ the $y$-value will be $3$. $(0,3)$

With this method you could calculate $y$-values and sketch the graph of the function.

**3rd step**: sketch the function

**4th step**: read the set of range from the graph: As you can see $y$-values starts from $3$. Therefore the range of $f\left(x\right)$ is $y\ge 3$.

**GDC**

In case you are allowed to use your calculator, graph the function to get a general understanding of what the function looks like. Once you have it, just find any special points or lines which determine the boundaries of the range

**Use your GDC to draw graphs of the following function: $f\left(x\right)={3}^{1-x}$**

**1st step**: In the menu go to the graphing mode and type the $f\left(x\right)={3}^{1-x}$ and press the ‘enter’ button.

**2nd step**: GDC will draw the function and now you are able to read the range and the domain of the function.

**Help**

The maximum and minimum points help you to define the range of the function. In some cases the minimum/maximum point will be the beginning or endpoint of the interval of the range. For example in case of quadratic functions.

**Find the range of the $f\left(x\right)={x}^{2}+1$, if the domain is $x\in \mathrm{\mathbb{R}}$!**

As we know the shape of a quadratic function we could sketch it. As you can see $y$-values starts from $1$. Therefore the range of $f\left(x\right)$ is $y\ge 1$.