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Answer :
Sure! Let's analyze each function and determine if its inverse is a function, and if so, find the inverse.
### 33. [tex]\( f(x) = 8x^3 \)[/tex]
The function [tex]\( f(x) = 8x^3 \)[/tex] is a cubic function. Cubic functions are one-to-one, meaning they have inverses that are also functions.
To find the inverse:
1. Start with [tex]\( y = 8x^3 \)[/tex].
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \left(\frac{y}{8}\right)^{1/3} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \left(\frac{x}{8}\right)^{1/3} \][/tex]
### 34. [tex]\( f(x) = -x^2 + 3 \)[/tex]
This function is a parabola opening downwards. Parabolic functions do not have inverses that are functions over all values of [tex]\( x \)[/tex] because they are not one-to-one.
Thus, the inverse of this function is not a function.
### 35. [tex]\( f(x) = x^3 + 4 \)[/tex]
The function [tex]\( f(x) = x^3 + 4 \)[/tex] is another cubic function. Like the previous cubic example, this function has an inverse that is also a function.
To find the inverse:
1. Start with [tex]\( y = x^3 + 4 \)[/tex].
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ y - 4 = x^3 \][/tex]
[tex]\[ x = (y - 4)^{1/3} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = (x - 4)^{1/3} \][/tex]
### 36. [tex]\( f(x) = 9x^2, \, x \geq 0 \)[/tex]
The function [tex]\( f(x) = 9x^2 \)[/tex] with the domain [tex]\( x \geq 0 \)[/tex] is a quadratic function with a restricted domain, making it one-to-one over its domain.
To find the inverse:
1. Start with [tex]\( y = 9x^2 \)[/tex].
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{\frac{y}{9}} \][/tex]
Since [tex]\( x \geq 0 \)[/tex], we only take the positive square root.
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \sqrt{\frac{x}{9}}, \, x \geq 0 \][/tex]
These are the steps to determine the inverses of the functions in the list. Each function is considered individually to see if its inverse can also be a function.
### 33. [tex]\( f(x) = 8x^3 \)[/tex]
The function [tex]\( f(x) = 8x^3 \)[/tex] is a cubic function. Cubic functions are one-to-one, meaning they have inverses that are also functions.
To find the inverse:
1. Start with [tex]\( y = 8x^3 \)[/tex].
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \left(\frac{y}{8}\right)^{1/3} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \left(\frac{x}{8}\right)^{1/3} \][/tex]
### 34. [tex]\( f(x) = -x^2 + 3 \)[/tex]
This function is a parabola opening downwards. Parabolic functions do not have inverses that are functions over all values of [tex]\( x \)[/tex] because they are not one-to-one.
Thus, the inverse of this function is not a function.
### 35. [tex]\( f(x) = x^3 + 4 \)[/tex]
The function [tex]\( f(x) = x^3 + 4 \)[/tex] is another cubic function. Like the previous cubic example, this function has an inverse that is also a function.
To find the inverse:
1. Start with [tex]\( y = x^3 + 4 \)[/tex].
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ y - 4 = x^3 \][/tex]
[tex]\[ x = (y - 4)^{1/3} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = (x - 4)^{1/3} \][/tex]
### 36. [tex]\( f(x) = 9x^2, \, x \geq 0 \)[/tex]
The function [tex]\( f(x) = 9x^2 \)[/tex] with the domain [tex]\( x \geq 0 \)[/tex] is a quadratic function with a restricted domain, making it one-to-one over its domain.
To find the inverse:
1. Start with [tex]\( y = 9x^2 \)[/tex].
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{\frac{y}{9}} \][/tex]
Since [tex]\( x \geq 0 \)[/tex], we only take the positive square root.
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \sqrt{\frac{x}{9}}, \, x \geq 0 \][/tex]
These are the steps to determine the inverses of the functions in the list. Each function is considered individually to see if its inverse can also be a function.
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