if f(x) = 2x^2+4 and g(x)=x-3 which number satisfies f(x)= (f o g)(x)
A) 3/2
B) 3/4
C) 5
D) 4
I am positive that answer A is correct, i just cant get the work to prove it, i can disprove all other answers and my teacher has assured me i am on the right track, he wants work proving answer A is correct and i have tried many different options such as substitution and even plugging in the variable to solve and i cant seem to do it right at all, i get incorrect and answers that don't make sense, i would really appreciate the help because i truly don't understand
(fâŚg)(x) = f(g(x))
Since f(x) = 2x^2+4,
f(g) = 2g^2+4 = 2(x-3)^2+4
So, we want
f(x) = (fâŚg)(x)
2x^2+4 = 2(x-3)^2 + 4
x^2 = (x-3)^2
x^2 = x^2-6x+9
0 = -6x+9
x = 3/2
Well, solving this problem can be a bit tricky, but fear not, I'm here to help you with a dash of humor!
To find the number that satisfies f(x) = (f o g)(x), we need to substitute g(x) into f(x). So let's have a clownish adventure!
First, let's substitute g(x) into f(x). That means we replace every x in f(x) with g(x) = x - 3, like so:
f(g(x)) = 2(g(x))^2 + 4
f(g(x)) = 2(x - 3)^2 + 4
Now, expand and simplify the equation:
f(g(x)) = 2(x^2 - 6x + 9) + 4
f(g(x)) = 2x^2 - 12x + 18 + 4
f(g(x)) = 2x^2 - 12x + 22
Okay, we're halfway there! We now have an expression for f(g(x)).
To find the answer, we need to find the x that makes f(x) equal to f(g(x)). In other words, we need to solve the equation:
f(x) = f(g(x))
2x^2 + 4 = 2x^2 - 12x + 22
Subtract 2x^2 from both sides:
4 = -12x + 22
Subtract 22 from both sides:
-18 = -12x
Divide by -12:
x = 3/2
Hooray! The correct answer is indeed A) 3/2! I hope my clownish explanation brought some laughter to your math journey. Keep up the good work!
To determine which number satisfies f(x) = (f o g)(x), we need to find the value of x that makes the equations equal.
First, let's find the expression for (f o g)(x) by substituting g(x) into f(x):
(f o g)(x) = f(g(x)) = f(x - 3) = 2(x - 3)^2 + 4 = 2(x^2 - 6x + 9) + 4 = 2x^2 - 12x + 18 + 4 = 2x^2 - 12x + 22
Now let's set f(x) equal to (f o g)(x):
2x^2 + 4 = 2x^2 - 12x + 22
To solve this equation, we can subtract 2x^2 from both sides to eliminate the common terms:
4 = -12x + 22
Then, we can subtract 22 from both sides:
-18 = -12x
Now, divide both sides by -12 to solve for x:
-18/-12 = x
x = 3/2
So, the number that satisfies f(x) = (f o g)(x) is x = 3/2, which corresponds to answer choice A.
To determine which number satisfies the equation f(x) = (f o g)(x), we need to substitute the expression for g(x) into f(x) and find the value of x that makes the equation true.
Let's start by substituting g(x) into f(x):
f(x) = 2x^2 + 4
(f o g)(x) = f(g(x)) = f(x-3) = 2(x-3)^2 + 4
Now we can set up the equation:
2x^2 + 4 = 2(x-3)^2 + 4
To solve this equation, we'll simplify and then equate the two sides:
2x^2 + 4 = 2(x^2 - 6x + 9) + 4
2x^2 + 4 = 2x^2 - 12x + 18 + 4
2x^2 + 4 = 2x^2 - 12x + 22
Next, we subtract 2x^2 from both sides of the equation to eliminate the x^2 terms:
4 = -12x + 22
To isolate the x term, we'll subtract 22 from both sides:
-18 = -12x
Finally, we divide by -12 to solve for x:
x = (-18)/(-12) = 3/2
Therefore, the number that satisfies f(x) = (f o g)(x) is x = 3/2, which corresponds to answer choice A.