2 The argument of a function

The input to a function is sometimes called its argument . It is frequently necessary to obtain the output from a function if we are given its argument. For example, given the function $g\left(t\right)=3t+2$ we may require the value of the output when the argument is 4. We write this as $g\left(t=4\right)$ or more usually and compactly as $g\left(4\right)$ . In this case the value of $g\left(4\right)$ is $3\times 4+2=14$ .

Example 2

Given the function $f\left(x\right)=3x+1$ find

1. $f\left(2\right)$
2. $f\left(-1\right)$
3. $f\left(6\right)$

Solution

1. The output from the function needs to be found when the argument or input is 2. We need to replace $x$ by 2 in the expression for the function. We find

   $f\left(2\right)=3\times 2+1=7$

2. Here the argument is $-1$ . We find

   $f\left(-1\right)=3\times \left(-1\right)+1=-2$

3. $f\left(6\right)=3\times 6+1=19$ .

Task!

It is possible to obtain the value of a function when the argument is an algebraic expression. Consider the following Example.

Example 3

Given the function $y\left(x\right)=3x+2$ find

1. $y\left(t\right)$
2. $y\left(2t\right)$
3. $y\left(z+2\right)$
4. $y\left(5x\right)$

Solution

The rule for this function is ‘multiply the input by 3 and then add 2’. We can apply this rule whatever the argument.

1. In this case the argument is $t$ . Multiplying this by 3 and adding 2 we find $y\left(t\right)=3t+2$ . Equivalently we can replace $x$ by $t$ in the expression for the function, so, $y\left(t\right)=3t+2$ .
2. In this case the argument is $2t$ . We need to replace $x$ by $2t$ in the expression for the function. So $y\left(2t\right)=3\left(2t\right)+2=6t+2$
3. In this case the argument is $z+2$ . We find $y\left(z+2\right)=3\left(z+2\right)+2=3z+8$ . It is important to note that $y\left(z+2\right)$ is not $y\times \left(z+2\right)=yz+y2$ but instead reads ‘ $y$ of $\left(z+2\right)$ ’ where ‘of’ means ‘take the function of’.
4. Here we have a complication. The argument is $5x$ and so there appears to be a clash of notation with the original expression for the function. There is no problem if we remember that the rule is to multiply the input by 3 and then add 2. The input now is $5x$ . So $y\left(5x\right)=3\left(5x\right)+2=15x+2$ .

Task!

Interactive Exercises

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