1. Find functions f and g so that f[g(x)]=H(x) where H(x)=(1+x^2)^3
2. For f(x)=4x+1 and g(x)=x^2-2x-4, find (g-f)(x)
3. Determine f(x)=x/x^2-4
4. Determine the domain of f in f(x)=10-sqrt3-x
1. To find functions f and g such that f[g(x)] = H(x) where H(x) = (1+x^2)^3, we can start by considering different compositions of functions. Let's break it down step by step:
Step 1: Determine g(x)
Start by finding a suitable function g(x) that would allow us to obtain (1+x^2)^3 when applied inside another function. In this case, one choice for g(x) could be g(x) = 1 + x^2. This function will give us (1+x^2) when evaluated.
Step 2: Determine f(y)
Next, we need to find a function f(y) that will produce our desired output (1+x^2)^3 when y = g(x). To achieve this, we can simply raise y to the power of 3, i.e., f(y) = y^3.
Step 3: Compose f and g
Finally, we substitute g(x) into f(y) to get our desired function f[g(x)]. Therefore, f[g(x)] = (1+x^2)^3.
2. To find (g - f)(x) for f(x) = 4x + 1 and g(x) = x^2 - 2x - 4, we need to subtract f(x) from g(x).
(g - f)(x) = g(x) - f(x)
Plugging in the given functions:
(g - f)(x) = (x^2 - 2x - 4) - (4x + 1)
Now simplify:
(g - f)(x) = x^2 - 2x - 4 - 4x - 1
(g - f)(x) = x^2 - 6x - 5
So, (g - f)(x) = x^2 - 6x - 5.
3. To determine the function f(x) = x/x^2 - 4, we can simplify the expression as follows:
f(x) = x / (x^2 - 4)
The denominator, x^2 - 4, can be factored as (x + 2)(x - 2).
Now, our function becomes:
f(x) = x / ((x + 2)(x - 2))
Therefore, the function f(x) = x / (x^2 - 4).
4. To determine the domain of the function f(x) = 10 - sqrt(3 - x), we need to consider the restrictions on x that would make the expression valid.
In this case, we want to avoid any values of x that would result in taking the square root of a negative number, as it is not defined in the real number system.
For the expression under the square root, 3 - x, to be non-negative, we must have:
3 - x ≥ 0
Solving for x, we get:
3 ≥ x
This means that x must be less than or equal to 3.
However, we also need to consider the denominator in the expression, which should not be equal to zero.
For f(x) to be defined, we must have:
x^2 - 4 ≠ 0
This implies that x cannot be ±2, as they would result in a zero denominator.
In summary, the domain of the function f(x) = 10 - sqrt(3 - x) is (-∞, -2) ∪ (-2, 2) ∪ (2, 3].