A Function of #x# means there is a "rule" (or process) which uses values of x and after some calculation, gives a y value.
#f(x) = y#
#color(magenta)(x ----rarr)# (some process) #color(magenta)( ---rarr y)#
The process is given as the expression in the function.
So, for #f(color(red)(x)) = color(red)(x)-2# we can find #f(4)# by substituting 4 for #x#
#f(color(red)(4)) = color(red)(4)-2 = 2 rarr# This gives # (x,y) " as "(4,2)#
Using different x-values will give a whole set of points.
The set of points is the function, which can also be graphed.
#g(x)# just means different "rule" or equation is used to find a y-value for any x-value that is used.
We have two functions. #f(x) " and " g(x)#
#f(x) = x-2 " and " g(x) = x^2 -7x-9#
Find #color(red)(f)(color(blue)(g(-1)))# means:
First use -1 for #x# in the 'g' rule, then use THAT value in the 'f' rule.
#color(magenta)(x ----rarr)# (some process) #color(magenta)( ---rarr y)#
#color(magenta)((-1) ----rarr (x^2 -7x-9) color(magenta)( ---rarr y)#
#color(blue)(g(-1)) " simply means use -1 for every "color(blue)(x)" in "color(blue)(x)^2-7color(blue)(x)-9#
#g(-1) = (-1)^2 -7(-1)-9#
# = 1 + 7 - 9#
# = color(red)(-1)#
Now find #color(red)(f(-1))#
#f(color(red)(-1)) = color(red)(-1)-2 = -3 #
