The parent function can be determined by direct substitution of the x values in the table into the expressions of each function. Otherwise, we could also just plot the points given in the table on a graph and observe the form of the graph, and this will help us choose the right option.
By Substitution method:
a) Given that f(x) = x^2
[tex]\begin{gathered} \text{ when x =-}2 \\ f(x)=(-2)^2=4 \end{gathered}[/tex]
This tallies with the y value in the table.
Again:
[tex]\begin{gathered} \text{ when x = -1} \\ f(x)=(-1)^2=1 \end{gathered}[/tex]
Since the y values tally with that given in the table, we can conclude that the parent function is f(x) = x^2
b) Given that f(x) = 2^x
[tex]\begin{gathered} \text{ when x = -2} \\ f(x)=2^{(-2)}=\frac{1}{4} \end{gathered}[/tex]
This does not tally with the y value in the table
c) Given that f(x) = |x|
[tex]\begin{gathered} \text{when x = -2} \\ f(x)=\lvert-2\rvert=2 \end{gathered}[/tex]
This does not tally with the y value in the table
d) Given that f(x) = x
[tex]\begin{gathered} \text{when x = -2} \\ f(x)=-2 \end{gathered}[/tex]
This also does not tally with the y value in the table
By Graphical method:
A plot of the values given in the table gives the following graph:
The above graph shows a parabola, which is obtained from quadratic functions.
Again, this points us to the conclusion that the parent function is f(x) = x^2
Thus, the answer is: option A
[tex]f(x)=x^2[/tex]
