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Anónimo preguntado en Science & MathematicsMathematics · hace 1 mes

# Function inverses?

How you get his answer to a problem like this? ### 3 respuestas

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• If f(g(x)) = g(f(x)) = x   f and g are inverses of each other

If f(g(x)) =  g(f(x)) ≠ x  f and g are not inverse of each other.

f(g(x)) = 4x - 3 and g(f(x)) = 4x + 3 which they are not equal to x, So they are not inverses of each other..

• An inverse of a function "undoes" whatever the function did.

If you give a value of 'x' to a function it gives 'y'.

Now if you feed 'y' into the *inverse* function, it should give you back 'x'.

For example, let's say a function multiplies a value by 4.

f(x) = 4x

Then the inverse function would need to *divide* the result by 4.

g(x) = x/4

So if you start with 10, you multiply by 4 and get 40, then when you use the inverse function, you divide by 4 to get back to 10.

It also works in reverse. If you start with the inverse function then apply the main function, you get back to the starting value. So again, if we start by 10 and *divide* by 4, we get 2.5. Then when we multiply by 4, we get back to 10.

Using symbols, if f(x) and g(x) are inverse functions then:

f(g(x)) = x

and

g(f(x)) = x

If you don't see this, then f(x) and g(x) are *not* inverse functions.

Clearly f and g in your problem are NOT inverses of each other.

We can test this with a value, like 10.

f(10) = 2(10) + 3 = 23

Now see if g(23) gives you 10:

g(23) = 2(23) - 3 = 43

Let's think what the inverse of f(x) = 2x + 3 should be.

We are multiplying by 2 and then adding 3. So the inverse function should *subtract* 3 and then *divide* by 2.

g(x) = (x - 3)/2

Let's test these two functions and show they do give back the original value.

f(10) = 2(10) + 3 = 23

g(23) = (23 - 3) / 2 = 10

And if you go in the other order:

g(10) = (10 - 3)/2 = 3.5

f(3.5) = 2(3.5) + 3 = 10

• if functions are inverse then

f(g(x))  =   x

so f and g are not inverses

the inverse of f(x) = 2x+3

replace x with f^(-1)(x) and solve for f^(-1)(x)

and replace f(x) = x

x =  2*f^(-1)(x) + 3

2*f^(-1)(x)  = x-3

f^(-1)(x)  =  (1/2)x  - (3/2)

so if

g(x) = (1/2)x -(3/2)  it would be the inverse

checking

f(g(x)) =  f( (1/2)x -(3/2)) =   2( (1/2)x -(3/2))  + 3

f(g(x)) =   x -3 + 3 = x

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