If f(x)=sin(x) and g(x)=x^2-x+1, find the domain of f(g(x)) and g(f(x)).
I was able to determine that:
f(g(x))= sin(x^2-x+1) and
g(f(x))= sin(x)^2 -sin(x) +1
Here is what I am thinking:
the range of f(x) is (x e R) and the range of g(x) is {x e R). Therefore the domain of f(g(x)) and g(f(x)) is {x e R)
the range of f(x) is {-1=< y=< 1} and the range of g(x) is {y>= .75}. Therefore the range of g(x) and f(x) will be what satisfies both the functions, { .75=< y =< 1}
Is this correct? If not, how would I go about solving this problem, specifically the range? Thanks for your help.
f(g(x))= sin(x^2-x+1) and
g(f(x))= sin(x)^2 -sin(x) +1
are correct.
The range of f(x)=sin(x) is [-1,1].
The range of g(x)=x²-x+1 is indeed ℝ.
However, the domain of both functions is ℝ.
So the domain of f(g(x)) ℝ and the range is still [-1,1].
The domain of g(f(x)) is ℝ. However, since sin(x) is a periodic function, we just have to limit the search to [-2π,2π].
Find the absolute minimum and absolute maximum on the interval [-2π,2π] and that would be the range (all possible values of the function).
To find the domain of a composite function like f(g(x)) or g(f(x)), we need to consider the restrictions imposed by both functions.
Let's start with f(g(x)) = sin(x^2 - x + 1):
1. The domain of g(x) is all real numbers since there are no restrictions on x.
2. However, inside the sine function, the argument (x^2 - x + 1) should be within the domain of the sine function, which is all real numbers. This means there are no restrictions on x for f(g(x)).
So, the domain of f(g(x)) is all real numbers.
Now let's consider g(f(x)) = (sin(x))^2 - sin(x) + 1:
1. The domain of f(x) is all real numbers since sine function accepts any real number as input.
2. Inside g(x), (sin(x))^2 is always non-negative because squaring any real number results in a non-negative value. There is no restriction on sin(x).
3. The subtraction of sin(x) and addition of 1 in g(x) do not impose any additional restrictions on the domain.
So, the domain of g(f(x)) is all real numbers as well.
Now, let's move on to the range of each function.
For f(x) = sin(x), the range is [-1, 1] since sine function values range from -1 to 1.
For g(x) = x^2 - x + 1, the range is determined by finding the vertex of the parabolic function. The vertex occurs at x = -b/2a = 1/2, and when we substitute this value into g(x), we get g(1/2) = 3/4. Since the coefficient of x^2 is positive, the parabola opens upwards, and the range is all y-values greater than or equal to 3/4, in other words, [3/4, ∞).
The range of f(g(x)) will be the range of f(x) intersected with the range of g(x), which is the interval [3/4,1].
Similarly, the range of g(f(x)) will also be the interval [3/4,1].
So, the range of f(g(x)) and g(f(x)) are both [3/4, 1].
To summarize:
- The domain of f(g(x)) is all real numbers.
- The domain of g(f(x)) is all real numbers.
- The range of f(g(x)) and g(f(x)) is [3/4, 1].