Calc Problem Functions?

Question

Let h(x) = 4 x2 + 4 x + 5.
Let g(x) = 2√{x} + 3.

what is
* g(x+1)
* h(h(x))
* h(g(x))
* g(h(x))

i keep getting these wrong, help

Answer 1

I'm going to assume that 4 x2 + 4 x + 5 means 4(x^2) + 4x + 5

The key is to subsitute the x in the original formula for WHATEVER is in the parenthesis next to the g or the h. If what's in the parenthesis is x, then all you have is the original formula.

g(x+1) = 2sqrt[x-1] + 3

h(h(x)) = 4(h(x))^2 + 4(h(x)) + 5 = 4(4x^2+4x+5)^2 + 4(4x^2+4x+5) + 5 = 64x^4 + 128x^3 + 160x^2 + 160x + 125

h(g(x)) = 4(g(x))^2 + 4(g(x)) + 5 = 4(2sqrt[x]+3)^2 + 4(2sqrt[x]+3) + 5 = 16x + 56sqrt[x] + 53

g(h(x)) = 2sqrt[h(x)] + 3 = 2sqrt[4(x^2) + 4x + 5] + 3 = 2sqrt[4((x^2) + x + 5/4] + 3 = 4sqrt[(x^2) + x + 5/4] + 3

The last two steps for g(h(x)) are optional. I threw them in because I though it looked a bit neater.

I sincerely hope this helped you. It seems far more complicated than it really is.

Always remember, before perfect clarity comes total confusion.

Answer 2

First, you have to understand what the above 2 functions mean.
g(x) refers to g function of x and similarly for h(x).

So g(x+1) means that the g function is applied to x+1, so in place of x, we substitute (x+1).
Hence,
g(x+1)=2√(x+1) + 3

As for composite functions like h(h(x)), h(g(x)), g(h(x)), it's rather similar. Shall you an example of 1 and you should be able to handle the rest. Oh yeah, and instead of writing h(h(x)), you can actually write it as h²(x), similarly hg(x) and gh(x) are understood.

For h(g(x)),

subtitute the function g(x), with h still outside.
hg(x)=h(2√{x} + 3)

Then substitute function h, where (2√{x} + 3) replaces x in h(x).
hg(x)=4[(2√{x} + 3)]² + 4[(2√{x} + 3)]+5

Simplify the expression and you should be able to get the ans
16x+56√(x)+53.

Try the other 2 qns, shall provide you with the final ans I calculated, see if you can get the hang of it.
h²(x)=224x^4 + 128x^3 + 80x² + 96x + 125
gh(x)=2√(4x²+4x+5) + 3

Hope this clears your doubts.

Answer 3

g(x+1) PLUG IN (X+1) for all x.
g(x+1) = 2√(x+1) + 3

For h(h(x))
Plug in 4 x2 + 4 x + 5 for all x.
h(h(x)) = 4[4 x2 + 4 x + 5]^2 + [4(4 x2 + 4 x + 5)] + 5
Do the same for the rest. It will be a little hard to simplify, though.

Answer 4

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Answer 5

g(x+1) = 2√{x+1} + 3

h(h(x)) = 4(4x² + 4x + 5)² + 4(4x² + 4x + 5) + 5
h(h(x)) = 4(16x^4 + 32x^3 + 56x² + 40x + 25) + 16x² + 16x + 20 + 5
h(h(x)) = 64x^4 + 128x^3 + 240x² + 176x +125

h(g(x)) = 4( 2√x + 3)² + 4( 2√x + 3) + 5
h(g(x)) = 4(4x +12√x + 9) + 8√x + 12 + 5
h(g(x)) = 16x + 56√x + 53

g(h(x)) = 2√(4x² + 4x + 5) + 3