b Replace f(x) with y. Then, switch x and y and solve for y.
C
c The domain of the inverse is the range of the function. The range of the inverse is the domain of the function.
A
a
B
b f^(- 1)(x) = (2(x+1))^2 - 4
C
cDomain: x ≥ -1
Range: y ≥ -4
Practice makes perfect
a We will first graph the original function. Let's recall its equation.
f(x) = sqrt(x+4)/2-1
To graph f(x), we will find some points that f(x) passes through. We can do so by finding the domain of f(x). Looking at the equation of f(x), the square root is the only part that adds a condition on the domain — the argument of a square root should be non-negative.
x+4 ≥ 0 ⇒ x ≥ -4The domain of f(x) is the set of x that are greater than or equal to -4. Let's substitute some values of x that are greater than or equal to -4 into the formula for f(x) and simplify.
x
sqrt(x+4)/2-1
(x, f(x))
-4
sqrt(-4+4)/2-1= -1
( -4, -1)
-3
sqrt(-3+4)/2-1= -0.5
( -3, -0.5)
0
sqrt(0+4)/2-1= 0
( 0, 0)
5
sqrt(5+4)/2-1= 0.5
( 5, 0.5)
Next, let's plot these points.
The last step for plotting the original function is to connect the points with a smooth curve. Let's go!
We can graph the inverse of the function by reflecting the graph of the original function across the line y=x. Let's add this line to our graph!
The next step is to reflect the graph of f(x) over the line y = x. To do so, we can first swap the coordinates of the points we plotted. The resulting points lie on the graph of the inverse function. If we connect them with a smooth curve, that curve should look like a reflection of the original function across that line.
Finally, let's take a look at the graph where only the function and the inverse are present.
b We want to find the inverse of the following function.
f(x) = sqrt(x+4)/2-1
First, we replace f(x) with y. From there, we switch x and y and solve for y.
y = sqrt(x+4)/2-1
⇓
x = sqrt(y+4)/2-1
The resulting equation will be the inverse of the given function.
Finally, to indicate that this is the inverse equation of f(x), we will replace y with f^(- 1)(x).
f^(- 1)(x) = (2(x+1))^2 - 4
c We want to find the domain and range of the inverse function f^(-1)(x). We will first find the domain and range of the original function. For this reason, let's recall the formula for f(x).
f(x) = sqrt(x+4)/2 - 1
In Part A, we found that the domain of f(x) is the set of those x that are greater than or equal to -4.
rc
Domain off(x):& x ≥ -4Now, let's recall the graph of the original function we made in Part A.
We see that f(x) is an increasing function, so its smallest value is for x = -4 — the smallest value of x within the domain. Let's find it!
The smallest value of f(x) is -1. Since the values of a square root are not bounded from above, the same is true for f(x). For this reason, the range of f(x) consists of those values of y, that are greater than -1.
rc
Domain off(x):& x ≥ -4
Range off(x):& y ≥ -1
Finally, recall that the domain of the inverse is the range of the original function and the range of the inverse is the domain of the original function. For this reason, let's swap the domain and the range of the original function to find the domain and range of the inverse f^(-1)(x).
rc
Domain off^(-1)(x):& x ≥ -1
Range off^(-1)(x):& y ≥ -4
