Functions: If f(x)=3x^2 - 4 and g(x) = x + 3, 1) Evaluate f(3), 2) Find the inverse of f(x) (f^-1(x)), 3)Find fg(x).
- in this case x=3 so you substitute it into f(x) to get f(3) = 3(3)2 - 4 = 3*9 -4 = 27 - 4 = 232) There are 2 methods to this question, a logical method, and a more mathematical method. The logical method is working backwards through BIDMAS so the last operation you perform with f(x) is -4, so the opposite of this would be the first operation of f-1(x) and the next is the 3 then the x2 , so f-1(x) = ((x+4)/3)1/2.The mathematical method is to make f(x)=y and rewrite the equation y=3x2 -4, then swap the two variable x & y, x=3y2-4, and finally you make y the subject of the equation by rearranging the equation. x+4=3y2 , (x+4)/3 = y2 , ((x+4)/3)1/2 = y, and therefore f-1(x) = ((x+4)/3)1/2, which is the same as the previous method, and a good way to check these answers is to put your answer from 1) into the equation and you should end up with 3, f-1(23) = ((23+4)/3)1/2=3.3)The hardest part is remembering the order in which you place the functions, the best way to think about it is working outwards from the brackets, so g is applied to x first, then once that function is complete, you apply f to the answer to that i.e. g(x) = x + 3. so fg(x) is the same as f(x+3) which is fg(x) = 3(x+3)2-4 and then you can expand this expression to get a quadratic equation, fg(x) = 3x2+18x + 23.
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