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Answer :
Certainly! Let's look at each function and determine whether the inverse is a function. When finding the inverse of a function, we switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the function equation, and then try to solve for [tex]\( y \)[/tex].
For a function [tex]\( f(x) \)[/tex] to have an inverse that is also a function, each [tex]\( x \)[/tex]-value must map to a unique [tex]\( y \)[/tex]-value (and vice versa). This means the original function should be one-to-one.
Here's a step-by-step analysis for each function:
33. [tex]\( f(x) = 8x^3 \)[/tex]
- Since cubic functions are one-to-one, this function does have an inverse. However, solving for the inverse is complex, so it's often deemed "No inverse" (not in elementary terms).
34. [tex]\( f(x) = -x^2 + 3 \)[/tex]
- This is a quadratic function and is symmetric around its vertex, which means it is not one-to-one. Hence, it does not have an inverse that is a function.
35. [tex]\( f(x) = x^3 + 4 \)[/tex]
- Similar to #33, cubic functions like this one are one-to-one, but finding a straightforward expression for the inverse can be complex.
36. [tex]\( f(x) = 9x^2, \, x \geq 0 \)[/tex]
- Even though this is a quadratic function, by restricting the domain to [tex]\( x \geq 0 \)[/tex], it becomes one-to-one and has an inverse.
37. [tex]\( f(x) = \frac{1}{4}x^2 \)[/tex]
- This function is not one-to-one over the entire real numbers, so it does not have an inverse that is a function.
38. [tex]\( f(x) = \frac{1}{5}x^5 \)[/tex]
- As with other odd power functions, this is one-to-one, but finding the practical inverse expression can be cumbersome.
39. [tex]\( f(x) = 2x^2 - 3 \)[/tex]
- This is a quadratic function, and without any domain restrictions, it is not one-to-one, so no inverse exists.
40. [tex]\( f(x) = x^4, \, x \geq 0 \)[/tex]
- Similar to #36, since [tex]\( x \geq 0 \)[/tex], it's one-to-one and an inverse exists.
41. [tex]\( f(x) = 5 - x^3 \)[/tex]
- This cubic function is one-to-one, but the inverse expression itself is complex to establish.
42. [tex]\( f(x) = x^5 - 2 \)[/tex]
- A one-to-one function as it's an odd degree polynomial, but again, inverses can be complex to express explicitly.
43. [tex]\( f(x) = x^5 + 1 \)[/tex]
- Like #42, it's one-to-one but the inverse requires complex expressions.
44. [tex]\( f(x) = 3x^2 - 4 \)[/tex]
- This quadratic function is not one-to-one over all real numbers, hence it does not have an inverse.
So, based on this analysis:
- Functions like quadratics without domain restrictions (e.g., #34, #37, #39, #44) do not have inverses that are functions.
- Certain polynomial functions like cubic functions or those with restricted domains do have inverses, but finding a simple expression for these inverses can often be tricky or impractical with basic algebra.
I hope this step-by-step approach clarifies which functions have inverses and why!
For a function [tex]\( f(x) \)[/tex] to have an inverse that is also a function, each [tex]\( x \)[/tex]-value must map to a unique [tex]\( y \)[/tex]-value (and vice versa). This means the original function should be one-to-one.
Here's a step-by-step analysis for each function:
33. [tex]\( f(x) = 8x^3 \)[/tex]
- Since cubic functions are one-to-one, this function does have an inverse. However, solving for the inverse is complex, so it's often deemed "No inverse" (not in elementary terms).
34. [tex]\( f(x) = -x^2 + 3 \)[/tex]
- This is a quadratic function and is symmetric around its vertex, which means it is not one-to-one. Hence, it does not have an inverse that is a function.
35. [tex]\( f(x) = x^3 + 4 \)[/tex]
- Similar to #33, cubic functions like this one are one-to-one, but finding a straightforward expression for the inverse can be complex.
36. [tex]\( f(x) = 9x^2, \, x \geq 0 \)[/tex]
- Even though this is a quadratic function, by restricting the domain to [tex]\( x \geq 0 \)[/tex], it becomes one-to-one and has an inverse.
37. [tex]\( f(x) = \frac{1}{4}x^2 \)[/tex]
- This function is not one-to-one over the entire real numbers, so it does not have an inverse that is a function.
38. [tex]\( f(x) = \frac{1}{5}x^5 \)[/tex]
- As with other odd power functions, this is one-to-one, but finding the practical inverse expression can be cumbersome.
39. [tex]\( f(x) = 2x^2 - 3 \)[/tex]
- This is a quadratic function, and without any domain restrictions, it is not one-to-one, so no inverse exists.
40. [tex]\( f(x) = x^4, \, x \geq 0 \)[/tex]
- Similar to #36, since [tex]\( x \geq 0 \)[/tex], it's one-to-one and an inverse exists.
41. [tex]\( f(x) = 5 - x^3 \)[/tex]
- This cubic function is one-to-one, but the inverse expression itself is complex to establish.
42. [tex]\( f(x) = x^5 - 2 \)[/tex]
- A one-to-one function as it's an odd degree polynomial, but again, inverses can be complex to express explicitly.
43. [tex]\( f(x) = x^5 + 1 \)[/tex]
- Like #42, it's one-to-one but the inverse requires complex expressions.
44. [tex]\( f(x) = 3x^2 - 4 \)[/tex]
- This quadratic function is not one-to-one over all real numbers, hence it does not have an inverse.
So, based on this analysis:
- Functions like quadratics without domain restrictions (e.g., #34, #37, #39, #44) do not have inverses that are functions.
- Certain polynomial functions like cubic functions or those with restricted domains do have inverses, but finding a simple expression for these inverses can often be tricky or impractical with basic algebra.
I hope this step-by-step approach clarifies which functions have inverses and why!
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