The one-to-one function f is defined by f(x)=(4x-1)/(x+7).
Find f^-1, the inverse of f. Then, give the domain and range of f^-1 using interval notation.
f^-1(x)=
Domain (f^-1)=
Range (f^-1)=
Any help is greatly appreciated.
Algebra - helper, Wednesday, February 2, 2011 at 7:05pm
f(x)=(4x-1)/(x+7)
y = (4x-1)/(x+7)
Rewrite as:
y = (4x)/(x+7)- 1/(x+7)
Multiply both sides by x+7:
(x + 7)y = 4x - 1
Expand out terms of the left hand side:
xy + 7y = 4x - 1
xy - 4x = -7y - 1
x(y - 4) = -7y - 1
Divide both sides by y - 4:
x = (-7y - 1)/(y - 4)
f^-1 = (-7x - 1)/(x - 4)
Can you do the domain and range now?
Algebra - Rachal, Wednesday, February 2, 2011 at 7:11pm
I don't know if this is right but this is what I came up with.
f^-1=(-7x+1)/(x-4)
domain f(^-1)=(-inf,-7)U(-7,inf)
range f(^-1)=(-inf,4)U(4,inf)
Let me know if it looks right. Thanks
The domain and range suggested apply to f(x). You will see that the vertical asymptote is at x=-7 when the denominator becomes zero.
f(x)=(4x-1)/(x+7)
domain f(^-1)=(-inf,-7)U(-7,inf)
range f(^-1)=(-inf,4)U(4,inf)
The domain and range of f-1(x) is equal to the range and domain respectively of f(x). Double check by evaluating the denominator at the singular points.
Post again if you need confirmation.
To find the inverse of a function, you need to switch the roles of x and y and solve for y. Let's follow the steps given in the previous answer to find f^(-1):
Given:
f(x) = (4x-1)/(x+7)
Step 1: Rewrite the equation with y instead of f(x):
y = (4x)/(x+7) - 1/(x+7)
Step 2: Switch x and y:
x = (4y)/(y+7) - 1/(y+7)
Step 3: Multiply both sides by (y+7):
x(y+7) = 4y - 1
Step 4: Expand out the left-hand side:
xy + 7x = 4y - 1
Step 5: Move all terms involving y to one side:
xy - 4y = -7x - 1
Step 6: Factor out y on the left-hand side:
y(x - 4) = -7x - 1
Step 7: Divide both sides by (x-4):
y = (-7x - 1)/(x - 4)
So, the inverse function f^(-1) is given by:
f^(-1)(x) = (-7x - 1)/(x - 4)
Now let's determine the domain and range of f^(-1) using interval notation.
Domain (f^-1):
The domain of f^(-1) is the set of all x-values for which the expression (-7x - 1)/(x - 4) is defined. The only restriction is that the denominator (x - 4) cannot be zero, so x ≠ 4. Therefore, the domain of f^(-1) is (-∞,4) U (4,∞).
Range (f^-1):
The range of f^(-1) is the set of all y-values that the function can take. Since the function is a rational function, the range can be any real number except for the values that make the denominator zero. Therefore, the range of f^(-1) is (-∞,∞).
So, the final answers are:
f^(-1)(x) = (-7x - 1)/(x - 4)
Domain (f^(-1)) = (-∞,4) U (4,∞)
Range (f^(-1)) = (-∞,∞)