f(x)=(x+7)/(x+3)
(f(x)-f(1))/(x-1)
To find the expression (f(x)-f(1))/(x-1), we start by substituting f(x) into the expression:
(f(x)-f(1))/(x-1) = ((x+7)/(x+3) - f(1))/(x-1)
Next, we need to find f(1) by substituting x=1 into the function f(x):
f(1) = (1+7)/(1+3) = 8/4 = 2
Substituting f(1) back into the expression:
((x+7)/(x+3) - 2)/(x-1)
Simplifying the expression further is possible, but as it is, this is the final result.
To find the step-by-step solution for [(f(x) - f(1))/(x - 1)], where f(x) = (x + 7)/(x + 3), follow these steps:
Step 1: Find f(1):
Substitute x = 1 into the function f(x) = (x + 7)/(x + 3) to find f(1):
f(1) = (1 + 7)/(1 + 3)
= 8/4
= 2
Step 2: Simplify (f(x) - f(1)):
Substitute f(1) = 2 into the function f(x) = (x + 7)/(x + 3):
(f(x) - f(1)) = (x + 7)/(x + 3) - 2
Step 3: Find the common denominator:
Multiply the first fraction [(x + 7)/(x + 3)] by (x + 3)/(x + 3) to create a common denominator:
(f(x) - f(1)) = [(x + 7)/(x + 3)] - 2(x + 3)/(x + 3)
Step 4: Combine the fractions:
(f(x) - f(1)) = [(x + 7 - 2(x + 3))]/(x + 3)
= (x + 7 - 2x - 6)/(x + 3)
= (-x + 1)/(x + 3)
Step 5: Simplify the expression (f(x) - f(1))/(x - 1):
[(f(x) - f(1))/(x - 1)] = (-x + 1)/(x + 3) divided by (x - 1)
Step 6: Invert the divisor and multiply:
[(f(x) - f(1))/(x - 1)] = (-x + 1)/(x + 3) * 1/(x - 1)
Step 7: Simplify the expression:
[(f(x) - f(1))/(x - 1)] = (-x + 1)/(x + 3)/(x - 1)
= (-x + 1)/[(x + 3)(x - 1)]
Therefore, the step-by-step solution for [(f(x) - f(1))/(x - 1)] is (-x + 1)/[(x + 3)(x - 1)].
To evaluate the expression (f(x) - f(1))/(x - 1), we first need to find the values of f(x) and f(1), and then substitute them into the expression.
Given the function f(x) = (x + 7)/(x + 3), let's find the values of f(x) and f(1):
1. To find f(x), substitute x into the function:
f(x) = (x + 7)/(x + 3)
2. To find f(1), substitute 1 into the function:
f(1) = (1 + 7)/(1 + 3)
Simplifying f(1), we have:
f(1) = 8/4
f(1) = 2
Now we have all the required values. Let's substitute them into the expression (f(x) - f(1))/(x - 1):
(f(x) - f(1))/(x - 1) = ((x + 7)/(x + 3) - 2)/(x - 1)
Simplifying the expression further, we can expand the numerator:
= ((x + 7 - 2(x + 3)) / (x + 3))/(x - 1)
= (x + 7 - 2x - 6) / (x + 3) / (x - 1)
= (-x + 1) / (x + 3) / (x - 1)
Therefore, the expression (f(x) - f(1))/(x - 1) simplifies to (-x + 1) / (x + 3) / (x - 1).