Question

If f(t)=t^3 +2, g(x)=2x+3, p(x)=(f(g(x)), and r(t)=f(g(t)). Compute p(-1) and r(-2).

Answer 1

to get p(-1), we first have to work out p(x),

p(x)=f(g(x))

what do you get when you substitute the definition of g(x) into this

Answer 2

p(x)=f( )

Answer 3

Okay I'm not sure what you mean, still new to functions

Answer 4

yeah i understand , the notation is a bit different to stuff you have learnt before,

ill give you a different example, and hopefully you can see what im doing

Answer 5

say h(x)= 100x ,
this is exactly the same as h(y)=100y
and also exactly the same as h(z)=100z

the function h is just multiplying what ever is in the brackets by 100,

so you could even say that h(cat)= 100cat

Answer 6

Okay, I follow

Answer 7

now say that j(x)= 55+ x

[same thing as j(cat)= 55 +cat ]

Answer 8

now we can put a more complicated thing in the bracket of the function
lets put the function h as the input for j

j(h(x))=j(100x)

now applying the definition of the j function ,
we get

j(100x)= 55+(100x)

Answer 9

or we could even put the function j as the input to the function j

j(j(t))= 55 + j(t)
= 55 + (55+t)
= 110 + t

am i making sense?

Answer 10

Yes, so far so good

Answer 11

cool,

now we can even use numbers as inputs


j(j(9))= 55+j(9)
= 55+55+9
=119

Answer 12

______________
______________



now if you can follow all this you should be able to
express p(x) [p as a function of x]

f(t) = t^3 +2,
g(x) = 2x+3,

p(x) = f(g(x))
= f( )
=

Answer 13

f(2x+3)

Answer 14

good, now apply the definition of the f function
(cube the input, and add two)

Answer 15

8+5?

Answer 16

should still have x's in it

Answer 17

[note that you dont have to simplify the cubed term leave it in brackets]

Answer 18

2x +5 (t^3)?

Answer 19

f(t) = t^3 +2,
g(x) = 2x+3,

p(x) = f(g(x))
= f(2x+3)
= ( )^3+2

Answer 20

p(x)=(2x+3)^3 +2

Answer 21

right, so you have an expression for p as a function x

now we can compute p(-1)

just put the -1 as the input for this definition of p

and then we can simplify the numbers

Answer 22

P(-1)= (-2 + -3)^3 +2
=-123

Answer 23

that would right [if p(x)=(2x-3)^3 +2]
but its a +3 [not -3]
p(x)=(2x+3)^3 +2

Answer 24

P(-1) =

Answer 25

oops 1 second

Answer 26

P(-1) = 3

Answer 27

YES!! very good √

Answer 28

Thanks a lot!

Answer 29

Is the r similar?

Answer 30

For r(-2). yes its the same idea ,

f(t) = t^3 +2
g(x) = 2x+3,

r(t) = f(g(t))
=
=

r(-2) =

Answer 31

is g(x) equivalent to g(t)?

Answer 32

r(-2)=(2x-3)^3 +2?

Answer 33

g(x) is equivalent to g(t)

Answer 34

so then f(2x+3) = (2x+3)^3 +2

Answer 35

yes
actually you can see that p and r are the same function

Answer 36

Oh okay

Answer 37

so you just need to get r(-2)=p(-2)

and you already worked out p(x)

Answer 38

so r(-2) = 1

Answer 39

The important thing to take away from this is that any time you have a function, we just call it a random letter like f or g or whatever and then in parenthesis you put another letter showing what you can replace in that function.

f(x) just means the function's name is f and anywhere you see an x in the function, you replace it.

You can replace the x with anything. A number. Another letter. A whole other equation. Or maybe an elephant.

So if you have f(x)=x and you want to replace x with 2, then you have f(2)=2. Just replacing stuff. If you want to replace x with (x^2-1) then f(x^2-1)=(x^2-1).

Another example, suppose you have g(t)=t^2+5t and you want to replace t with elephants? No problem, g(elephants)=(elephants)^2+5(elephants).

Answer 40

Thanks that makes sense

Answer 41

+3 [not -3]

sorry, i have to go catch a train

if you still can't get this one ,
you might like to close this question and open a new one, just for r(-2)
and if you show your working somelse will be able to help

Answer 42

Okay, thanks for all your help