PLS PLS HELP ITS SOO URGENT

Question

So we have this table to as an example to help us throughout the questions

Student Pair	Function one	Function 2	Composition
1	g(x)=1/x	f(x)=(2x^2)-8	g(f(x))
2	k(x)=sqrt(x)	h(x)= x-1	k(h(x))
3	L(x)= |x|	m(x)= -x^2	L(m(x))
4	t(x)=2 ^ (x+1)	s(x)=log_2 (x)	t(s(x))

1.  Choose two of the pairs of functions and the associated composition. Complete the table for the given values of x.

x	Function 2	Composition Function1(Function2(x))
-3
-2
-1
0
1
2

2.Make a conjecture (your best guess) for the domain of your composition. Explain how you made this conjecture.

3.Make a conjecture for the range of your composition. Explain how you made this conjecture.

4.Write the algebraic representation or formula for this composition. Use this function formula to complete the table

x	Composition Function1(Function2(x))
-3
-2
-1
0
1
2
3

How do the values in this table compare to the table you completed in part 1?

5.Which table do think is easier to complete, the table in part 1 or in part 4? Explain why.

6.Use your graphing calculator to graph the composition you selected in part 1:

7.

a. Sketch the graph of this composition. How does this graph support or contradict the domain and range you identified in parts b and c?

b. Consider the functions w(x)=ln⁡(x−1) and y(x)=ex+1

c. Create the composition w(y(x)).

d. Graph the two functions w(x) and y(x)on the same screen. What do you notice about these two graphs?

Answer 1

When we think about composition functions remember that x is just a place holder for "some number." x can also be another function. You could also think of this as g(f(x)) = 1/(f(x)) = 1/((2x^2)-8).

So for the first row on the table we can find f(-3)= (2(-3)^2) - 8 = 10. f(x) is "some number" and at x=-3 that number is 10. So g(f(-3)) is the same as g(x) where x=10. So g(f(-3)) = g(10) = 1/10.

Domain just explains what the "some numbers" are that work for this function. For example, g(0) = 1/0 which is not allowed. So is there a way we can get 0=f(x)? What x-values would be needed? After looking at f(x) for a bit we find that there are no x-values that could make the function equal to zero. In fact, because of the x^2, the function cannot equal anything less than f(0)= -8. If f(x) has a lower domain of -8 than g(f(x)) must as well. Try putting increasingly larger/smaller numbers in for x and see what happens to the function. With a composition function, the function g(x) must be dependent on f(x) and therefore that function's x-values. The range, similarly, is the function's possible y-values.

With the graph, you should be able to see if the composition function uses x or y-values out to infinity, or if there are some numbers that the function cannot use/produce. For the function w(y(x)) you can use the same techniques.

I hope that helps. Please let me know if you have any other questions or would like some more in-depth/specific explanations.

Answer 2

For completing the tables, simply choose one function 1 and one function 2. For the 2nd column of the table in pt 1, simply plug the given x-values into whichever function 2 you chose and write the corresponding y-values you get from that function 2. Then, to complete the 3rd column, plug the values from the second column in for x into the function 1 you chose. (This is one way to get points for a composition function. You plug x into the inside function, take whatever y that outputs and put it in for x to the outside function to get the y-value for the entire composition. You use the first x you plugged in as x, the last value you got out as y. The middle number doesn't matter.)

You are instructed to make guesses for what the domain is for this composition (what are the x-values that generate defined y-values). Same thing for range.

To write the algebraic representation, simply write plug in the expression on the right-hand side of the equation for Function 2 in for x in the equation for function 1. For example, if you chose the functions in row 2, the equation would be k(h(x)) = sqrt(x-1).

For number 7, the 2 given functions are inverses, so their composition should give w(y(x)) = x for all x.