We've seen the form "$y=$y=" when we learnt about graphing. We've used this form to describe straight lines, parabolas and hyperbolas just to name a few.
In Is this a Functional Relationship, we were also introduced to the concept of functions. Where each input yielded a unique output.
When we are writing in function notation, instead of writing "$y=$y=", we write "$f(x)=$f(x)=". This gives us a bit more flexibility when we're working with equations or graphing as we don't have to keep track of so many $y$ys! Instead, using function notation, we can write $f(x)=$f(x)=, $g(x)=$g(x)=, $h(x)=$h(x)= and so on. These are all different expressions that involve only $x$x as the variable.
We can also evaluate "$f(x)$f(x)" by substituting values into the equations just like we would if the question was in the form "$y=$y=".
Examples
Question 1
If $A(x)=x^2+1$A(x)=x2+1 and $Q(x)=x^2+9x$Q(x)=x2+9x, find:
A) $A(5)$A(5)
Think: This means we need to substitute $5$5 in for $x$x in the $A(x)$A(x) equation.
Do:
| $A(5)$A(5) | $=$= | $5^2+1$52+1 |
| $=$= | $26$26 |
B) $Q(6)$Q(6)
Think: This means we need to substitute $6$6 in for $x$x in the $Q(x)$Q(x) equation.
Do:
| $Q(6)$Q(6) | $=$= | $6^2+9\times6$62+9×6 |
| $=$= | $36+54$36+54 | |
| $=$= | $90$90 |
C) $A(10)+Q(7)$A(10)+Q(7)
Think: This question will use both equations.
Do:
Let's work it out separately first:
| $A(10)$A(10) | $=$= | $10^2+1$102+1 |
| $=$= | $101$101 | |
| $Q(7)$Q(7) | $=$= | $7^2+9\times7$72+9×7 |
| $=$= | $112$112 | |
| $101+112$101+112 | $=$= | $213$213 |
We can also do this as one long calculation:
| $10^2+1+7^2+9\times7$102+1+72+9×7 | $=$= | $101+112$101+112 |
| $=$= | $213$213 |
Question 2
Consider the function $f\left(x\right)=8x+6$f(x)=8x+6.
Determine the output produced by the input value $x=-5$x=−5.
Question 3
If $Z(y)=y^2+12y+32$Z(y)=y2+12y+32, find $y$y when $Z(y)=-3$Z(y)=−3.
Write both solutions on the same line separated by a comma.