Given f(x)=5+6x-x^2 find and simplify.
A)f(t-2)=____?
B)f(z+h)-f(z)=____?
C)f(b)+3=_____?
To find and simplify the given expressions, substitute the corresponding values into the function f(x) = 5 + 6x - x^2 and simplify the resulting expressions.
A) To find f(t-2), substitute t-2 in place of x:
f(t-2) = 5 + 6(t-2) - (t-2)^2
Now simplify the expression:
f(t-2) = 5 + 6t - 12 - (t^2 - 4t + 4)
Combine like terms:
f(t-2) = -7 + 6t - t^2 + 4t - 4
Simplify further by rearranging the terms:
f(t-2) = -t^2 + 10t - 11
So, f(t-2) simplifies to -t^2 + 10t - 11.
B) To find f(z+h) - f(z), substitute z+h and z into the function:
f(z+h) - f(z) = [5 + 6(z+h) - (z+h)^2] - [5 + 6z - z^2]
Now simplify the expression:
f(z+h) - f(z) = 5 + 6z + 6h - (z^2 + 2zh + h^2) - 5 - 6z + z^2
Combine like terms:
f(z+h) - f(z) = 6h - 2zh - h^2
Simplify further by rearranging the terms:
f(z+h) - f(z) = -h^2 - 2zh + 6h
So, f(z+h) - f(z) simplifies to -h^2 - 2zh + 6h.
C) To find f(b) + 3, substitute b in place of x:
f(b) + 3 = (5 + 6b - b^2) + 3
Now simplify the expression:
f(b) + 3 = 5 + 6b - b^2 + 3
Combine like terms:
f(b) + 3 = 8 + 6b - b^2
Rearrange the terms for simplification:
f(b) + 3 = -b^2 + 6b + 8
So, f(b) + 3 simplifies to -b^2 + 6b + 8.