Answer:
1. Option D is correct.
2. Option B is correct.
3. Option B is correct.
Step-by-step explanation:
Inverse function defined as the the function that undergoes the action of the other function.
A function
is the inverse of f if whenever y =f(x) and ![x =f^{-1}](https://tex.z-dn.net/?f=x%20%3Df%5E%7B-1%7D)
To find the inverse of the function:
Q1.
Given the function: f(x) = 7x -1
Put y for f(x) and solve for x;
y= 7x -1
Add 1 both sides we get;
y + 1 = 7x
Divide both sides by 7 we get;
![x = \frac{y+1}{7}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7By%2B1%7D%7B7%7D)
Put
for x;
![f^{-1}(y) = \frac{y+1}{7}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28y%29%20%3D%20%5Cfrac%7By%2B1%7D%7B7%7D)
Interchange y =x, we have
Q 2.
Given the function:
![f(x) = x^3 - 7](https://tex.z-dn.net/?f=f%28x%29%20%3D%20x%5E3%20-%207)
Put y for f(x) and solve for x;
![y = x^3-7](https://tex.z-dn.net/?f=y%20%3D%20x%5E3-7)
Add 7 both sides we get;
![y + 7 =x^3](https://tex.z-dn.net/?f=y%20%2B%207%20%3Dx%5E3)
taking cube root both sides we get
![x =\sqrt[3]{y+7}](https://tex.z-dn.net/?f=x%20%3D%5Csqrt%5B3%5D%7By%2B7%7D)
Put
for x;
![f^{-1}(y) =\sqrt[3]{y+7}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28y%29%20%3D%5Csqrt%5B3%5D%7By%2B7%7D)
Interchange y =x, we have
![f^{-1}(x) = \sqrt[3]{x+7}](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%20%3D%20%5Csqrt%5B3%5D%7Bx%2B7%7D)
Q3 .
Given the function:
![f(x) = 5x^3 - 3](https://tex.z-dn.net/?f=f%28x%29%20%3D%205x%5E3%20-%203)
Put y for f(x) and solve for x;
![y = 5x^3-3](https://tex.z-dn.net/?f=y%20%3D%205x%5E3-3)
Add 3 both sides we get;
![y + 3 =5x^3](https://tex.z-dn.net/?f=y%20%2B%203%20%3D5x%5E3)
Divide both sides by 5 we get;
![x^3 = \frac{y+3}{5}](https://tex.z-dn.net/?f=x%5E3%20%3D%20%5Cfrac%7By%2B3%7D%7B5%7D)
taking cube root both sides we get
![x = \sqrt[3]{\frac{y+3}{5} }](https://tex.z-dn.net/?f=x%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7By%2B3%7D%7B5%7D%20%7D)
Put
for x;
![f^{-1}(y) = \sqrt[3]{\frac{y+3}{5} }](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28y%29%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7By%2B3%7D%7B5%7D%20%7D)
Interchange y =x, we have
![f^{-1}(x) = \sqrt[3]{\frac{x+3}{5} }](https://tex.z-dn.net/?f=f%5E%7B-1%7D%28x%29%20%3D%20%5Csqrt%5B3%5D%7B%5Cfrac%7Bx%2B3%7D%7B5%7D%20%7D)