When we solve for the inverse of a function, we just simply interchange x and y and then equate the equation to y. So the original function, f(x), has an inverse function, f^-1(x), where the domain of the original function is the range of its inverse while the range of the original function is the domain of its inverse. Now let us try to obtain the inverse function of the following given functions.
1. f(x) = ½x + 4
First step is to change f(x) to y.
So: y = ½x + 4
Now let us change x to y and y to x. This will become:
x = ½y + 4
With algebraic manipulation, we will equate the equation to y:
2(x = ½y + 4)
2x = y + 8
Then we transpose 8 to the left side of the equation to isolate y. So:
2x – 8 = y
which can also be written as:
y = 2x – 8
We can also factor out the terms to simplify further:
y = 2 (x–4)
And change y to f^-1(x).
Therefore, the inverse function of f(x) = ½x + 4 is f^-1(x) = 2 (x–4).
2. f(x) = (4x–1)/(2x+1)
y = (4x–1)/(2x+1)
x = (4y–1)/(2y+1)
x(2y+1) = 4y–1
2xy+x = 4y–1
2xy+x–4y = –1
2xy–4y = –x–1
We will factor out the terms at the left.
y(2x–4) = –x–1
y = (–x–1)/(2x–4)
y = –(x+1)/(2x–4)
f^-1(x) = –(x+1)/(2x–4)
Therefore, the inverse function of f(x) = (4x–1)/(2x+1) is f^-1(x) = –(x+1)/(2x–4).
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SOLVING FOR THE INVERSE FUNCTION
Answer with Step-by-step explanation:
When we solve for the inverse of a function, we just simply interchange x and y and then equate the equation to y. So the original function, f(x), has an inverse function, f^-1(x), where the domain of the original function is the range of its inverse while the range of the original function is the domain of its inverse. Now let us try to obtain the inverse function of the following given functions.
1. f(x) = ½x + 4
First step is to change f(x) to y.
So: y = ½x + 4
Now let us change x to y and y to x. This will become:
x = ½y + 4
With algebraic manipulation, we will equate the equation to y:
2(x = ½y + 4)
2x = y + 8
Then we transpose 8 to the left side of the equation to isolate y. So:
2x – 8 = y
which can also be written as:
y = 2x – 8
We can also factor out the terms to simplify further:
y = 2 (x–4)
And change y to f^-1(x).
Therefore, the inverse function of f(x) = ½x + 4 is f^-1(x) = 2 (x–4).
2. f(x) = (4x–1)/(2x+1)
y = (4x–1)/(2x+1)
x = (4y–1)/(2y+1)
x(2y+1) = 4y–1
2xy+x = 4y–1
2xy+x–4y = –1
2xy–4y = –x–1
We will factor out the terms at the left.
y(2x–4) = –x–1
y = (–x–1)/(2x–4)
y = –(x+1)/(2x–4)
f^-1(x) = –(x+1)/(2x–4)
Therefore, the inverse function of f(x) = (4x–1)/(2x+1) is f^-1(x) = –(x+1)/(2x–4).
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