Answer :
Sure, let's analyze each given function step by step to determine which one has an inverse that is also a function.
1. Function [tex]\( b(x) = x^2 + 3 \)[/tex]:
- This is a quadratic function, which is represented by a parabola.
- A function has an inverse that is also a function if it is one-to-one; that is, it passes the horizontal line test.
- The graph of [tex]\( b(x) = x^2 + 3 \)[/tex] is a parabola that opens upwards.
- Since this parabola fails the horizontal line test (a horizontal line would intersect the parabola at two points), [tex]\( b(x) \)[/tex] is not one-to-one over all real numbers.
- Therefore, it does not have an inverse that is a function.
2. Function [tex]\( d(x) = -9 \)[/tex]:
- This is a constant function.
- For any input [tex]\( x \)[/tex], the output is always [tex]\( -9 \)[/tex].
- A constant function is not one-to-one because multiple x-values map to the same y-value.
- Since it fails the one-to-one criterion, it cannot have an inverse that is a function.
3. Function [tex]\( m(x) = -7x \)[/tex]:
- This is a linear function.
- Linear functions of the form [tex]\( y = mx + b \)[/tex] (where [tex]\( m \neq 0 \)[/tex]) are always one-to-one.
- For any unique [tex]\( x \)[/tex]-value, there is a unique [tex]\( y \)[/tex]-value, meaning the function passes the horizontal line test and is invertible.
- To find the inverse of the function:
[tex]\[ y = -7x \implies x = -\frac{y}{7} \implies m^{-1}(y) = -\frac{y}{7} \][/tex]
- Therefore, [tex]\( m(x) = -7x \)[/tex] has an inverse that is a function.
4. Function [tex]\( p(x) = |x| \)[/tex]:
- This is the absolute value function.
- The graph of [tex]\( p(x) = |x| \)[/tex] is V-shaped, opening upwards.
- It does not pass the horizontal line test because, for example, [tex]\( p(2) = 2 \)[/tex] and [tex]\( p(-2) = 2 \)[/tex]. Thus, it is not one-to-one.
- Consequently, it does not have an inverse that is a function.
Based on the detailed analysis, the function that has an inverse which is also a function is:
[tex]\( m(x) = -7x \)[/tex].
Thus, the function with an inverse that is a function corresponds to [tex]\( m(x) = -7x \)[/tex], which is the third option. The answer is:
[tex]\[ 3 \][/tex]
1. Function [tex]\( b(x) = x^2 + 3 \)[/tex]:
- This is a quadratic function, which is represented by a parabola.
- A function has an inverse that is also a function if it is one-to-one; that is, it passes the horizontal line test.
- The graph of [tex]\( b(x) = x^2 + 3 \)[/tex] is a parabola that opens upwards.
- Since this parabola fails the horizontal line test (a horizontal line would intersect the parabola at two points), [tex]\( b(x) \)[/tex] is not one-to-one over all real numbers.
- Therefore, it does not have an inverse that is a function.
2. Function [tex]\( d(x) = -9 \)[/tex]:
- This is a constant function.
- For any input [tex]\( x \)[/tex], the output is always [tex]\( -9 \)[/tex].
- A constant function is not one-to-one because multiple x-values map to the same y-value.
- Since it fails the one-to-one criterion, it cannot have an inverse that is a function.
3. Function [tex]\( m(x) = -7x \)[/tex]:
- This is a linear function.
- Linear functions of the form [tex]\( y = mx + b \)[/tex] (where [tex]\( m \neq 0 \)[/tex]) are always one-to-one.
- For any unique [tex]\( x \)[/tex]-value, there is a unique [tex]\( y \)[/tex]-value, meaning the function passes the horizontal line test and is invertible.
- To find the inverse of the function:
[tex]\[ y = -7x \implies x = -\frac{y}{7} \implies m^{-1}(y) = -\frac{y}{7} \][/tex]
- Therefore, [tex]\( m(x) = -7x \)[/tex] has an inverse that is a function.
4. Function [tex]\( p(x) = |x| \)[/tex]:
- This is the absolute value function.
- The graph of [tex]\( p(x) = |x| \)[/tex] is V-shaped, opening upwards.
- It does not pass the horizontal line test because, for example, [tex]\( p(2) = 2 \)[/tex] and [tex]\( p(-2) = 2 \)[/tex]. Thus, it is not one-to-one.
- Consequently, it does not have an inverse that is a function.
Based on the detailed analysis, the function that has an inverse which is also a function is:
[tex]\( m(x) = -7x \)[/tex].
Thus, the function with an inverse that is a function corresponds to [tex]\( m(x) = -7x \)[/tex], which is the third option. The answer is:
[tex]\[ 3 \][/tex]
