Answer:
A
Step-by-step explanation:
We have the two functions: [tex]f(x)=1-x^2[/tex] and [tex]g(x)=x^2-1[/tex].
And we want to find the value of b such that [tex]f(3b)+8=g(3b)-8[/tex] is true.
Notice that we can multiply f(x) by -1. This yields:
[tex]-f(x)=x^2-1[/tex]
Notice that this is the same as g(x). Therefore:
[tex]-f(x)=g(x)[/tex]
If we substitute 3b for x:
[tex]-f(3b)=g(3b)[/tex]
So, we can go back to our equation. We have:
[tex]f(3b)+8=g(3b)-8[/tex]
Substitute -f(3b) for g(3b):
[tex]f(3b)+8=-f(3b)-8[/tex]
Solve. Subtract 8 from both sides and add f(3b) to both sides:
[tex]2f(3b)=-16[/tex]
Divide both sides by 2:
[tex]f(3b)=-8[/tex]
Now, we can use the function of f(x). Substitute 3b into f(x). So:
[tex]1-(3b)^2=-8[/tex]
Subtract 1 from both sides:
[tex]-(3b)^2=-9[/tex]
Divide both sides by -1:
[tex](3b)^2=9[/tex]
Take the square root of both sides:
[tex]3b=\pm 3[/tex]
Divide both sides by 3. So, the value of b can be:
[tex]b=-1, 1[/tex]
Therefore, our answer is A.
And we're done!
