f(x)= x + 3. Find and simplify :
f(a+h)−f(a) h
the x +3 is under a radical sign as well! so its the square root of x +3
its also / h!
So we have f(x) = √(x+3)
and I bet it is also : (f(a+h)−f(a))/h
(f(a+h)−f(a))/h
= ( √(a+h + 3) - √(a+3) )/h
= ( √(a+h + 3) - √(a+3) )/h * (√(a+h + 3) + √(a+3) ) / (√(a+h + 3) + √(a+3) )
= (a+h+3 - a-3)/( h(√(a+h + 3) + √(a+3) ))
= h/( h(√(a+h + 3) + √(a+3) )
= 1/(√(a+h + 3) + √(a+3) ) , h ≠ 0
To find and simplify the expression f(a+h) - f(a) / h, we need to start by evaluating f(a+h) and f(a) individually.
Given that f(x) = x + 3, we substitute (a+h) and a into the function:
f(a+h) = (a+h) + 3
f(a) = a + 3
Now we can substitute these values into the expression f(a+h) - f(a) / h:
[(a+h) + 3 - (a + 3)] / h
Next, simplify the numerator by combining like terms:
[(a + h + 3) - (a + 3)] / h
(a + h + 3 - a - 3) / h
(a - a + h + 3 - 3) / h
(h) / h
Finally, simplify the expression by canceling out the h terms:
h / h = 1
Therefore, the simplified expression f(a+h) - f(a) / h is equal to 1.