Given that f(x) = x^2-3 and g(x) = 2x+1, find each of the following, if it exists.
a. (fg)(-root3)
b. (gf)(-1/2)
do you mean f(g(x)) ?
f(g(x))
= (g(x))^2 - 3
= (2x+1)^2 - 3
then (f(g(-√3))
= (-2√3 + 1)^2 - 3
= 12 - 4√3 + 1 - 3
= 10 - 4√3
do the other one the same way
remember this time you want
g(f(x))
= g(x^2-3)
= 2(x^2-3) + 1
The first one was (fg)(-root3). Does this answer your ? Thanks for the help!
What happend to the (-1/2) in the second problem?
To find the values of (fg)(-√3) and (gf)(-1/2), we need to simplify the composition of the functions f and g using the given values.
a. To find (fg)(-√3), we first need to evaluate g(-√3) and then substitute this result into the function f.
1. Evaluate g(-√3):
g(x) = 2x + 1
g(-√3) = 2(-√3) + 1
To simplify this expression, we multiply -√3 by 2 and then add 1:
g(-√3) = -2√3 + 1
2. Substitute the result g(-√3) into f(x):
f(x) = x^2 - 3
f(g(-√3)) = f(-2√3 + 1)
To simplify this expression, we substitute (-2√3 + 1) into the function f:
f(-2√3 + 1) = (-2√3 + 1)^2 - 3
Now, we simplify the expression further:
(-2√3 + 1)^2 = (-2√3 + 1)(-2√3 + 1)
= 4 * 3 - 2 * √3 * 2√3 + 1 * 1
= 12 - 12 + 1
= 1
Thus, (fg)(-√3) = (-2√3 + 1)^2 - 3 = 1 - 3 = -2
b. To find (gf)(-1/2), we need to evaluate f(-1/2) and then substitute this result into the function g.
1. Evaluate f(-1/2):
f(x) = x^2 - 3
f(-1/2) = (-1/2)^2 - 3
To simplify this expression, we square -1/2 first:
(-1/2)^2 = (-1/2) * (-1/2)
= 1/4
Next, we substitute 1/4 into the function f:
f(-1/2) = 1/4 - 3
Now, we simplify the expression further:
1/4 - 3 = 1/4 - 12/4
= -11/4
2. Substitute the result f(-1/2) into g(x):
g(x) = 2x + 1
g(f(-1/2)) = g(-11/4)
To simplify this expression, we substitute -11/4 into the function g:
g(-11/4) = 2(-11/4) + 1
To simplify further, we multiply -11/4 by 2 and then add 1:
g(-11/4) = -11/2 + 1
Now, we simplify the expression:
-11/2 + 1 = -11/2 + 2/2
= -9/2
Thus, (gf)(-1/2) = -9/2.