Find the inverse of the function.

Question

y(x)=2x+ 1/2

Answer 1

A Relation is a set of ordered pairs. If you swap the x and y coordinates of each ordered pair you get the Inverse Relation. The Inverse's graph is a reflection of the original Relation's graph over the line y = x.

 

If you have an equation like y = 2 x - 3 you swap the variables and solve for the new y: x = 2 y - 3, 2 y = x + 3, y = (x + 3)/2.

 

For your problem:

 

f(x) = y = 2x + 1/2 is the original equation.

 

Swapping the variables:

x = 2y + 1/2

 

Solving for y:

x - 1/2 = 2y

(x - 1/2)/2 = y

f^(-1)(x) = y = 1/2 x - 1/4 is the inverse equation

 

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Every Relation has an Inverse; one or both may also be Functions.

 

The inverse of the parabola y = x^2 is y = ±sqrt(x), which is not a function of x because there is more than one value of y for each x.

 

The inverse of the relation y = ±sqrt(x), which is not a function, is the parabola y = x^2, which is a function.

 

An origin centered circle, x^2 + y^2 = r^2, is its own inverse; neither are functions. Also any line through the origin is a line of symmetry of the circle. The perfect shape.

Answer 2

Hi Katelynn;

y(x)=2x + 1/2

I believe this is...

f(x)=2x + (1/2)

We need...

f^-1(x)=2x+(1/2)

Let's begin with

y=2x + (1/2)

Let's isolate x...

Let's subtract 1/2 from both sides...

(-1/2)+y=2x+ (1/2) - (1/2)

y-(1/2)=2x

Let's divide both sides by 2...

[y-(1/2)]/2=(2x)/2

(1/2)y-(1/4)=x

Now, let switch x and y...

(1/2)x-(1/4)=y

 

Both equations are in slope-intercept format...

y=mx+b

m is the slope.

b is the y-intercept, the value of y when x=0.

The first equation has a slope of 2 and a y-intercept of (0,1/2).

The second equation has a slope of (1/2) and a y-intercept of (0,-1/4).

The second equation is NOT perpendicular to the first.  To be perpendicular, it would have a negative slope as opposed to the positive slope of the original equation.  However, if you graph is, you will see that the two points meet at...

(1/2)x-(1/4)=2x+(1/2)

and that they are mirror images of each other.

 

Answer 3

Hi Katelynn,

 

The definition of an inverse function, f^-1(x) is f(f^-1(x))=x….in words, f inverse "undoes what f does.

 

so if f(x)=2x+1/2 then f^-1(x) = x/2 - 1/4 because f(f^-1(x)) =2*(x/2-1/4) +1/2 =x-1/2+1/2= x.

 

One way to calculate the inverse of a simple function is to solve for x as a function of y and then replace y with x. i,e, in your case y=2x+1/2 so x=(1/2)y - 1/4 and so f^-1(x)=(1/2)x -1/4.

 

Hope this is not too confusing

Jim