1. F(x)=x^2-5 and G(x)=1-x Find (F/G)(x)= 2. For the the following function, a) determine wheter it is one to one, b) if it is one to one, find its inverse function f(x)=x+9) Is the given function a one to one function? Select correct choice and fill in blank a. Yes, the function is one to one. The inverse function is f^-1(x)=_____ b. No, the function is not one to one 3. Determine wheter the function is one to one. If it is find a formula for its inverse. f(x)=9x-8 Yes or no 4. Determine whether the function is one to one. If it is find a formula for its inverse. f(x)=x^2-6 Is the function one to one? Select the correct choice below and fill in the blank a. Yes the function is one to one. The inverse function is f^-1(x)=_____ b. No, the function is not one to one Solution 1) F(x)=x^2-5 and G(x)=1-x (F/G)(x)= (x^2-5) / (1-x) This is not a one to one function 2) f(x) = x+9 yes this is a one to one function. All linear functions are one-to-one. inverse will be worked as follows: x=y+9 We change the f(x) to x, and the original x\'s to y\'s. x-9=y We then solve for y. f -1(x) = x-9 3) f(x)=9x-8 This is a linear function, so again, it is a one-to-one. f -1(x) = (x+8) / 9 4) f(x)=x^2-6 This is a parabola and thus cannot be a one-to-one function. Quick rule of thumb: If all the exponents of the variables are even numbered, a one-to-one function is not possible. If all the exponents of the variables are odd numbered, then it is a one- to-one function. Exceptions can appear when using rational functions such as the first example..

1. 1. Factor the polynomial 12 x6 -25x4+30x3+20 in Z[x] or prove it cannot be done.
2. Expand (x-2)2(x-4)(x-6) in Z8[x] and findall the roots in Z8 of the result.
Solution
2. Expand (x-2)2(x-4)(x-6) in Z8[x] and findall the roots in Z8 of the result. the
roots are x = 2, 4 ,6 (x-2)2(x-4)(x-6) = (x2- 4x +4)(x2 -10x +24) = x4 - 10x3+24x2 -4x3 + 40x2 -
96x +4x2 - 40x +96 = x4 - 14x3 + 68x2 -136x +96