The most common notation used for a function is $f(x)$f(x) . We can use any letters we wanted though: you might see $g(t)$g(t), $h(x)$h(x) or $p(z)$p(z) . There's no reason we can't use a capital letter either, such as $A(x)$A(x) or $B(t)$B(t). In the fields of engineering and science the letters are chosen according to the quantity they represent.
We saw function notation briefly when defining functions and relations, and it looks like this:
We read this as "$f$f as a function of $x$x equals $x$x squared".
Our input to the function goes into the brackets that follow the $f$f. So, if we want to find the value of the function when $x=3$x=3, we can write and evaluate this as $f(3)=3^2=9$f(3)=32=9.
As you can see, function notation gives us a shorthand for substitution. If $y=3x+2$y=3x+2 and we write this is function notation as $f(x)=3x+2$f(x)=3x+2. Then the question 'what is the value of $y$y when $x$x is $5$5?' can be asked simply as 'what is $f(5)$f(5)?' or 'evaluate $f(5)$f(5).' Remember, if you are given a graph of a function, 'evaluate $f(5)$f(5)' is asking you to find the $y$y value that corresponds to $x=5$x=5.
There's no reason to think of $x$x as only representing whole numbers. You may be asked to evaluate $f(x)$f(x) using fractions, surds or even an input of an algebraic term. We can also work backwards to find $x$x if we are given a value of $f(x)$f(x).
Practice Questions
Question 1
Consider the function $f\left(x\right)=2x^3+3x^2-4$f(x)=2x3+3x2−4.
Evaluate $f\left(0\right)$f(0).
Evaluate $f\left(\frac{1}{4}\right)$f(14).
Question 2
A function $f\left(x\right)$f(x) is defined by $f\left(x\right)=\left(x+4\right)\left(x^2-4\right)$f(x)=(x+4)(x2−4).
Evaluate $f\left(6\right)$f(6).
Find all solutions for which $f\left(x\right)=0$f(x)=0.
Question 3
Consider the function $f\left(x\right)=x^2-3x-2$f(x)=x2−3x−2.
Form an expression for $f\left(a-2\right)$f(a−2).
Form an expression for $f\left(a+h\right)-f\left(a\right)$f(a+h)−f(a).
Question 4
Use the graph of the function $f\left(x\right)$f(x) to find each of the following values.
$f\left(0\right)$f(0)
$f\left(-2\right)$f(−2)
Find the value of $x$x such that $f\left(x\right)=2$f(x)=2
Operations on functions
Functions can be added, subtracted, multiplied or divided as long as they exist for the same $x$x values. If $f$f and $g$g are functions, we compute $f(x)+g(x)$f(x)+g(x) by adding the function values at $x$x for each function. The short-hand notation for function addition is $(f+g)(x)$(f+g)(x). In a similar way, we form $(f-g)(x)$(f−g)(x) for subtraction and $(f/g)(x)$(f/g)(x) for function division. You might see function multiplication written in different ways such as $(f\times g)(x)$(f×g)(x) , $(f\cdot g)(x)$(f·g)(x)or just simply $(fg)(x)$(fg)(x).
Practice Questions
Question 5
Let $f\left(x\right)=-5x+3$f(x)=−5x+3 and $g\left(x\right)=x^2-7$g(x)=x2−7.
Find the function $\left(f\times g\right)\left(x\right)$(f×g)(x):
Evaluate $\left(f\times g\right)$(f×g)$\left(-3\right)$(−3):