Answer :
To determine which of the given functions have inverses that are also functions, let's analyze each of them one by one.
1. Function [tex]\( g(x) = 2x - 3 \)[/tex]:
- This is a linear function. Linear functions are bijective when the slope is non-zero (which it is in this case since the slope is 2). A bijective function has an inverse that is also a function.
- The inverse of [tex]\( g(x) \)[/tex] can be found as follows:
[tex]\[ y = 2x - 3 \implies x = \frac{y + 3}{2} \implies g^{-1}(y) = \frac{y + 3}{2} \][/tex]
- Thus, [tex]\( g(x) \)[/tex] has an inverse that is also a function.
2. Function [tex]\( k(x) = -9x^2 \)[/tex]:
- This is a quadratic function and is not one-to-one because for any positive value of [tex]\( x \)[/tex], [tex]\( k(x) \)[/tex] yields the same value as for the corresponding negative value of [tex]\( x \)[/tex]. Quadratic functions generally do not have inverses that are functions without restricting the domain.
- Since [tex]\( k(x) \)[/tex] is not one-to-one over all real numbers, it does not have an inverse that is a function.
3. Function [tex]\( f(x) = |x + 2| \)[/tex]:
- This is an absolute value function. Absolute value functions are not one-to-one because they map both [tex]\( x \)[/tex] and [tex]\( -x \)[/tex] to the same value.
- For instance, [tex]\( f(0) = |0 + 2| = 2 \)[/tex] and [tex]\( f(-4) = |-4 + 2| = 2 \)[/tex].
- As a result, [tex]\( f(x) \)[/tex] does not have an inverse that is a function.
4. Function [tex]\( w(x) = -20 \)[/tex]:
- This is a constant function, mapping every [tex]\( x \)[/tex] to the same value, -20. Constant functions are not bijective because they are not one-to-one.
- Therefore, [tex]\( w(x) \)[/tex] does not have an inverse that is a function.
Based on this analysis, only the function [tex]\( g(x) = 2x - 3 \)[/tex] has an inverse that is also a function.
1. Function [tex]\( g(x) = 2x - 3 \)[/tex]:
- This is a linear function. Linear functions are bijective when the slope is non-zero (which it is in this case since the slope is 2). A bijective function has an inverse that is also a function.
- The inverse of [tex]\( g(x) \)[/tex] can be found as follows:
[tex]\[ y = 2x - 3 \implies x = \frac{y + 3}{2} \implies g^{-1}(y) = \frac{y + 3}{2} \][/tex]
- Thus, [tex]\( g(x) \)[/tex] has an inverse that is also a function.
2. Function [tex]\( k(x) = -9x^2 \)[/tex]:
- This is a quadratic function and is not one-to-one because for any positive value of [tex]\( x \)[/tex], [tex]\( k(x) \)[/tex] yields the same value as for the corresponding negative value of [tex]\( x \)[/tex]. Quadratic functions generally do not have inverses that are functions without restricting the domain.
- Since [tex]\( k(x) \)[/tex] is not one-to-one over all real numbers, it does not have an inverse that is a function.
3. Function [tex]\( f(x) = |x + 2| \)[/tex]:
- This is an absolute value function. Absolute value functions are not one-to-one because they map both [tex]\( x \)[/tex] and [tex]\( -x \)[/tex] to the same value.
- For instance, [tex]\( f(0) = |0 + 2| = 2 \)[/tex] and [tex]\( f(-4) = |-4 + 2| = 2 \)[/tex].
- As a result, [tex]\( f(x) \)[/tex] does not have an inverse that is a function.
4. Function [tex]\( w(x) = -20 \)[/tex]:
- This is a constant function, mapping every [tex]\( x \)[/tex] to the same value, -20. Constant functions are not bijective because they are not one-to-one.
- Therefore, [tex]\( w(x) \)[/tex] does not have an inverse that is a function.
Based on this analysis, only the function [tex]\( g(x) = 2x - 3 \)[/tex] has an inverse that is also a function.
