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.

To find the domain and range of the composite function G(h(x)), we need to consider the individual domains and ranges of the functions G(x) = sin(x) and h(x) = x^2 - x + 1, as well as any restrictions imposed due to composition.

Let's start with the domain of h(x), which is the set of all possible input values for the function. Since h(x) is a polynomial, it is defined for all real numbers. Therefore, the domain of h(x) is (-∞, ∞).

Next, let's consider the range of h(x), which is the set of all possible output values. As a quadratic function, h(x) represents a parabola opening upward, and its vertex gives us the minimum value of the function. We can use calculus or complete the square to find that h(x) = x^2 - x + 1 has a minimum value of 3/4. Hence, the range of h(x) is [3/4, ∞).

Moving on to the domain of G(x) = sin(x), we know that the sine function is defined for all real numbers. Thus, the domain of G(x) is also (-∞, ∞).

The range of G(x) is the set of all possible output values. The sine function oscillates between -1 and 1. Therefore, the range of G(x) is [-1, 1].

Now, let's consider the composition G(h(x)). The composition G(h(x)) means that we apply the function G(x) to the output of h(x). In this case, that means G(h(x)) = sin(h(x)).

To find the domain of the composite function G(h(x)), we need to consider any potential restrictions or limitations introduced by the composition. In this case, there are no restrictions or limitations because both G(x) = sin(x) and h(x) = x^2 - x + 1 are defined for all real numbers.

Therefore, the domain of G(h(x)) is (-∞, ∞).

To find the range of the composite function G(h(x)), we need to evaluate G(h(x)) for all possible input values from the domain of h(x).

Since h(x) is defined for all real numbers, we can plug any real number into h(x) and then apply the sine function to that result. As mentioned before, the range of G(x) is [-1, 1], so the range of G(h(x)) will be a subset of [-1, 1].

However, finding the exact range of G(h(x)) can be challenging without further information or analysis. It may involve solving equations or using numerical methods.

Thus, for the composite function G(h(x)), the domain is (-∞, ∞), and the range is a subset of [-1, 1].