1.Give a quadratic function q(x) with a vertex of (-1, 2).

2. Give a function h(x) which does not have an inverse function.

3. Give a polynomial function g(x) with zeros 1,2,3

4. Give an exponential function s(x) passing through (0,2) and (4,1).

5. Give a rational function R(x) with a zero of 2 and a vertical asymptote of x=3.

q(x) = 73(x+1)^2+2

h(x) = x^2
g(x) = (x-1)(x-2)(x-3)

Since x^0=1 for any x, we have s(x) = 2*a^(kx) for some a and k.
2*1/2 = 1, so
s(x) = 2*4^(-x/2)

r(x) will involve something/(x-3)
x-2 is zero at x=2, so
r(x) = (x-2)/(x-3) will work

1. To find a quadratic function with a vertex of (-1, 2), we can take the general form of a quadratic function, which is q(x) = a(x - h)^2 + k, where (h, k) represents the vertex. In this case, h = -1 and k = 2. So, a possible quadratic function q(x) would be q(x) = a(x + 1)^2 + 2, where a is any non-zero real number.

2. A function h(x) that does not have an inverse function can be any non-one-to-one function. For example, a function like h(x) = x^2 is not one-to-one because multiple x values can produce the same output. In order for a function to have an inverse, it needs to be one-to-one, meaning each x value corresponds to a unique y value.

3. To find a polynomial function g(x) with zeros 1, 2, and 3, we can use the factored form of the polynomial. Since the zeros are given as 1, 2, and 3, we can construct the factored form as g(x) = a(x - 1)(x - 2)(x - 3), where a is any non-zero real number.

4. To find an exponential function s(x) passing through the points (0, 2) and (4, 1), we can use the general form of an exponential function, which is s(x) = a*b^x, where a is a non-zero constant and b is the base. By substituting the coordinates, we get the following equations: 2 = a * b^0 and 1 = a * b^4. From the first equation, we find that a = 2. Substituting this value in the second equation, we get 1 = 2 * b^4. Solving for b, we find that b = 1/2. Therefore, the exponential function s(x) passing through the given points is s(x) = 2 * (1/2)^x.

5. To find a rational function R(x) with a zero of 2 and a vertical asymptote of x=3, we can start by setting up the equation using the factored form of a rational function. A rational function can be written as R(x) = a(x - p)/(x - q), where p represents the zero(s) and q represents the vertical asymptote(s). In this case, the zero is given as 2, so we have (x - 2) in the numerator. The vertical asymptote is given as x=3, so we have (x - 3) in the denominator. Combining these, we get R(x) = a(x - 2)/(x - 3), where a is any non-zero real number.