I have to find the domain and range for the composite of functions G(x)=sinx and h(x)=x^2-x+1. I'm confused about this and would really appreciate some help. Thanks.
The composite of functions g(x)=sin(x) and h(x)=x²-x+1 can be represented by ("g" of "h")
g o h(x)
=g( h(x) )
=sin( h(x) )
=sin(x²-x+1)
The domain is limited by h(x) and g(x).
Since the domain of both h(x) and g(x) is ℝ, the domain of g o h(x) is also ℝ.
I will leave it to you to find the range.
Thank you!!!
f(x)=2/x+7,g(x)=28/x
To find the domain and range of the composite function, we need to determine the restrictions on both the input and output of the composed functions.
Let's first find the composite function G(h(x)). To do this, we substitute h(x) into G(x) and simplify:
G(h(x)) = sin(h(x)) = sin(x^2 - x + 1)
Now, let's consider the domain of the composite function. The domain is the set of all possible input values for which the function is defined. In this case, since both G(x)=sinx and h(x)=x^2 - x + 1 are defined for all real numbers, the composite function G(h(x)) is also defined for all real numbers.
Next, let's find the range of the composite function. The range is the set of all possible output values of the function. In this case, the range of sin(x) is [-1, 1]. However, when we compose sin(x) with another function, such as h(x), the range can change.
To find the range of G(h(x)), we need to analyze the behavior of h(x)=x^2 - x + 1. Since this is a quadratic function, its graph is a parabola. The vertex of the parabola occurs at x = -b/(2a), where a and b are coefficients in the quadratic equation.
For h(x) = x^2 - x + 1, a = 1 and b = -1. So the x-coordinate of the vertex is x = -(-1)/(2 * 1) = 1/2. Since the coefficient of x^2 is positive, the parabola opens upward, and the vertex (1/2, h(1/2)) is the minimum point on the graph.
Therefore, the range of h(x) is [h(1/2), ∞). To find h(1/2), substitute 1/2 into h(x):
h(1/2) = (1/2)^2 - (1/2) + 1 = 1/4 - 1/2 + 1 = 3/4
So the range of h(x) is [3/4, ∞).
Now we can determine the range of the composite function G(h(x)). Since sin(x) is bounded between -1 and 1, the range of G(h(x)) is simply equal to the range of sin(x), which is [-1, 1].
In summary, the domain of the composite function G(h(x)) is all real numbers, and the range is [-1, 1].