p(1) = 1
Let p(x) = x2 + bx = c
then, p(p(x)) = (x2 + bx + c)² + b(x2 + bx + c) + c
Sum of roots of p(p(x)) is given by = - (coefficient of x3/constant term)
= -2b/c2+bc+c = f(a,b) (say)
For critical points,
∂f/∂c = 0 and ∂f/∂b = 0
= 2b(2c+b+1)/c2+bc+c = 0 and - {( c2+bc+c)²- 2b (c)/ ( c2+bc+c)²}= 0
= b(2c+b+1) =0
= b= 0 or b= - (1+2c) = -(2c(c+1))/c2+bc+c = 0
If c=0 , b= -1 => c= 0 or c=-1
If c = -1 ,b = 1
thus we have the following possibility for (b,c)
(0,0) , (0,-1) , (-1,0) ,(1,-1).
At (0,0) f(0,0) = not defined
(0,-1) f(0,-1) = 0
(1,-1) f(1,-1) = 2
(-1,0) is not possible.
p(x) = (x2 +x-1)² + (x2 +x-1) -1
= p(1) = (1+1-1)² + (1+1-1) -1
= 1
Hence, p(1) = 1
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a) Increasing function.
b) The growth factor is 4.
c) The asymptote is y tending to zero as x tends to negative infinity.
d) y = 1
c) The graph can be seen at the end.
<h3>
What conclusions can we get about the given function?</h3>
Here we are given the following exponential function:
y =f(x) = 4^x
a) First, we want to know if the function is increasing or decreasing. Remember that a function is increasing if, as x increses, also does the value of y.
Here clearly we have an increasing function.
- f(0) = 4^0 = 1
- f(1) = 4^1 = 4
- f(2) = 4^2 = 16
and so on...
b) Now we want to identify the growth factor. This is the number that is being exponentiated, so in:
y = 4^x
The growth factor is 4.
c) We have an asymptote as x tends to negative infinity, because we will eventually have denominator that tends to infinity, then we will have an asymptote that tends to zero, as x tends to negative infinity.
d) The y-intercept is what we get when we evaluate in x = 0, we have:
y = f(0) = 4^0 = 1
The y-intercept is y = 1.
e) The sketch of the exponential function can be seen in the image below.
If you want to learn more about exponential functions:
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Answer:
5/36
Step-by-step explanation:
1 times 5 over 9 times 4
In mathematics, a rational number is a number such as −3/7 that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q