Answer :
To determine which function has an inverse that is also a function, we need to analyze each given option to check if they are one-to-one functions. A one-to-one function means that each x value maps to a unique y value and vice versa, which is necessary for the inverse to also be a function.
### Analysis of Each Function:
1. [tex]\( b(x) = x^2 + 3 \)[/tex]
- This is a quadratic function.
- Quadratic functions are not one-to-one because they fail the horizontal line test; that is, a horizontal line will intersect the graph at more than one point.
- Therefore, [tex]\( b(x) = x^2 + 3 \)[/tex] does not have an inverse that is a function.
2. [tex]\( d(x) = -9 \)[/tex]
- This is a constant function.
- Constant functions are also not one-to-one because any horizontal line test or multiple x-values map to the same y-value.
- Therefore, [tex]\( d(x) = -9 \)[/tex] does not have an inverse that is a function.
3. [tex]\( m(x) = -7x \)[/tex]
- This is a linear function of the form [tex]\( y = mx + c \)[/tex] where [tex]\( m \neq 0 \)[/tex].
- Linear functions are one-to-one; they pass the horizontal line test because each x-value maps to a unique y-value.
- Therefore, [tex]\( m(x) = -7x \)[/tex] has an inverse that is a function, which can be found by solving for x: [tex]\( x = \frac{y}{-7} \)[/tex].
4. [tex]\( p(x) = |x| \)[/tex]
- This is an absolute value function.
- Absolute value functions are not one-to-one because they fail the horizontal line test; for example, [tex]\( p(x) = 1 \)[/tex] when [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
- Therefore, [tex]\( p(x) = |x| \)[/tex] does not have an inverse that is a function.
### Conclusion:
After analyzing all the options, the function that has an inverse which is a function is:
- [tex]\( m(x) = -7x \)[/tex]
Therefore, the correct answer is:
3
### Analysis of Each Function:
1. [tex]\( b(x) = x^2 + 3 \)[/tex]
- This is a quadratic function.
- Quadratic functions are not one-to-one because they fail the horizontal line test; that is, a horizontal line will intersect the graph at more than one point.
- Therefore, [tex]\( b(x) = x^2 + 3 \)[/tex] does not have an inverse that is a function.
2. [tex]\( d(x) = -9 \)[/tex]
- This is a constant function.
- Constant functions are also not one-to-one because any horizontal line test or multiple x-values map to the same y-value.
- Therefore, [tex]\( d(x) = -9 \)[/tex] does not have an inverse that is a function.
3. [tex]\( m(x) = -7x \)[/tex]
- This is a linear function of the form [tex]\( y = mx + c \)[/tex] where [tex]\( m \neq 0 \)[/tex].
- Linear functions are one-to-one; they pass the horizontal line test because each x-value maps to a unique y-value.
- Therefore, [tex]\( m(x) = -7x \)[/tex] has an inverse that is a function, which can be found by solving for x: [tex]\( x = \frac{y}{-7} \)[/tex].
4. [tex]\( p(x) = |x| \)[/tex]
- This is an absolute value function.
- Absolute value functions are not one-to-one because they fail the horizontal line test; for example, [tex]\( p(x) = 1 \)[/tex] when [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
- Therefore, [tex]\( p(x) = |x| \)[/tex] does not have an inverse that is a function.
### Conclusion:
After analyzing all the options, the function that has an inverse which is a function is:
- [tex]\( m(x) = -7x \)[/tex]
Therefore, the correct answer is:
3
