Question:
If the expression (x^2 + 2x +4)/(x^2 - 2x - 4) has a value between (1/3) and 3 for all real values of x, determine the range of the expression [9.(3)^x + 6.(3)^2x + 4]/[ 9.(3)^2x - 6.(3)^x +4
My thoughts on the question:
I can see that if we substitute x=[3.(3)^x] we can turn the first expression into the one ,which we are told to find the range.But I have no idea on going further..
But (x^2 + 2x +4)/(x^2 - 2x - 4) does not always fit into the interval [1/3,3]. If x is very close to 1±√5 then the value is arbitrarily large
f(3.2) = -129
plus, you have a typo. I think you mean
[9.(3)^2x + 6.(3)^x + 4]/[ 9.(3)^2x - 6.(3)^x +4]
using u = 3.3^x
That becomes (u^2+2u+4)/(u^2-2u-4)
so, I guess its range is [1/3,3]
To determine the range of the expression [9.(3)^x + 6.(3)^(2x) + 4] / [9.(3)^(2x) - 6.(3)^x + 4], we can follow these steps:
1. Substitute x = 3.(3)^x into the given expression (x^2 + 2x + 4) / (x^2 - 2x - 4).
2. Simplify the expression using the substitution. We get:
[9.(3)^(3.(3)^x) + 6.(3)^(2.(3.(3)^x)) + 4] / [9.(3)^(2.(3.(3)^x)) - 6.(3)^(3.(3)^x) + 4].
3. Now, we need to determine the range of this new expression. To do this, we need to consider the range of the original expression (x^2 + 2x + 4) / (x^2 - 2x - 4).
4. We are given that the original expression has a value between 1/3 and 3 for all real values of x.
5. To find the range of the original expression, we can set it equal to 1/3 and 3 and solve for x. By doing this, we can find the range of x values that satisfy the given conditions.
6. After finding the range of x values, substitute these values into the new expression to determine the range of [9.(3)^x + 6.(3)^(2x) + 4] / [9.(3)^(2x) - 6.(3)^x + 4].
7. This will give us the final range of the expression.
I apologize if my previous response was not clear. Please follow these steps to determine the range of the given expression.