Given f, left bracket, x, right bracket, equals, 3, x, squared, plus, 2f(x)=3x
2
+2, find f, left bracket, minus, 7, right bracketf(−7)
To find f(-7), we plug -7 into the function f(x) = 3x^2 + 2f(x) + 3x + 2:
f(-7) = 3(-7)^2 + 2f(-7) + 3(-7) + 2
Simplifying the equation, we get:
f(-7) = 147 + 2f(-7) - 21 + 2
Rearranging the terms:
f(-7) - 2f(-7) = 147 - 19
Combining like terms:
-f(-7) = 128
Multiplying both sides by -1:
f(-7) = -128
Therefore, f(-7) is equal to -128.
To find f(-7) using the given equation f(x) = 3x^2 + 2f(x) = 3x + 2, we need to substitute -7 for x and solve for f(-7).
Step 1: Substitute -7 for x in the equation.
f(-7) = 3(-7)^2 + 2f(-7) = 3(49) + 2f(-7) = 147 + 2f(-7)
Step 2: Simplify the equation.
f(-7) = 147 + 2f(-7)
Step 3: Move 2f(-7) to the other side of the equation.
f(-7) - 2f(-7) = 147
Step 4: Combine like terms.
-f(-7) = 147
Step 5: Multiply both sides by -1 to isolate f(-7).
f(-7) = -147
So, f(-7) is equal to -147.
To find the value of f(-7), we first need to find the expression for f(x) using the given information.
We are given that f(x) equals 3x^2 + 2(f(x) = 3x^2 + 2).
And we also have the equation 2f(x) = 3x + 2.
Let's use the second equation to solve for f(x):
2f(x) = 3x + 2
Divide both sides of the equation by 2:
f(x) = (3x + 2) / 2
Now, we have an expression for f(x) which is (3x + 2) / 2.
To find f(-7), substitute -7 in place of x in the expression:
f(-7) = (3(-7) + 2) / 2
Now perform the calculations:
f(-7) = (-21 + 2) / 2
f(-7) = -19 / 2
f(-7) = -9.5
So f(-7) is equal to -9.5.