Jacob K. answered • 12/14/21
McGill Grad for Nighttime Math Tutoring and Emergency Help
Let's go through each of them and see what we can find out
f(x)=1-x^2,g(x)=x+5
A. f0g(x)=(1-x^2)(x+5)
f o g(x) means f composed of g(x), so essentially we take the expression of g(x), which ix x+5, and we insert it into f(x) where we see x. so we're putting in x+5 into f(x) which gives us 1-(x+5)^2. We can expand this and factor it out to get (x+5)(x+5)=x^2+10x+25, and the total expression then coming to 1-(x^2+10x+25). Let's now see if this is the same as (1-x^2)(x+5).
(1-x^2)(x+5)=x+5-x^3-5x^2. This is not equal to what we found the expression of f o g(x) to be. A is false.
B. f(g(-1))=-3
First, we find what g(-1) is by inserting -1 into g(x), giving us g(-1)=(-1)+5=4. We now put this value into f(x), getting f(4)=1-4^2=1-16=-15. -15=/=-3. This is false.
C. gof(2)=2
Similar to A, g o f(x) means we put f(x) into the expression of g(x). However, this time, we are given a value of x to begin with. f(x)=1-x^2, so then f(2) = 1-(2)^2=1-4=-3. So -3=f(2). g o f(2) = (1-(-3)^2)+5=(1-9)+5=-8+5=-3. -3=/=2.
D. f(g(x))=1+(x+5)^2
We just have to do f o g(x) here, which is 1 - (x+5)^2 . I believe that this is the correct answer, and that there is a typo in the problem as written. This is the closest to a correct answer there is.
I hope that these explainations help! Good luck
