To determine which of the given functions has an inverse that is also a function, we need to examine each one to see if it passes the horizontal line test. The horizontal line test states that if any horizontal line intersects the graph of a function at more than one point, the function does not have an inverse that is a function.
Let's analyze each function:
1. [tex]\( d(x) = -9 \)[/tex]:
- This is a constant function, which means that for any value of [tex]\( x \)[/tex], [tex]\( y \)[/tex] will always be -9.
- Graphically, this is a horizontal line.
- Therefore, it fails the horizontal line test because every horizontal line will intersect this function at infinitely many points.
- Conclusion: [tex]\( d(x) \)[/tex] does not have an inverse that is a function.
2. [tex]\( b(x) = x^2 + 3 \)[/tex]:
- This is a quadratic function.
- The graph is a parabola opening upwards with vertex at [tex]\((0, 3)\)[/tex].
- Any horizontal line above [tex]\( y = 3 \)[/tex] will intersect the parabola at two points.
- Therefore, it fails the horizontal line test.
- Conclusion: [tex]\( b(x) \)[/tex] does not have an inverse that is a function.
3. [tex]\( p(x) = |x| \)[/tex]:
- This is the absolute value function.
- The graph is a V-shaped curve with vertex at the origin.
- Any horizontal line above [tex]\( y = 0 \)[/tex] will intersect the absolute value function at two points.
- Therefore, it fails the horizontal line test.
- Conclusion: [tex]\( p(x) \)[/tex] does not have an inverse that is a function.
4. [tex]\( m(x) = -7x \)[/tex]:
- This is a linear function with a slope of -7.
- The graph is a straight line with a negative slope.
- Any horizontal line will intersect the graph at most at one point.
- Therefore, it passes the horizontal line test.
- Conclusion: [tex]\( m(x) \)[/tex] does have an inverse that is a function.
Thus, among the given functions, only [tex]\( m(x) = -7x \)[/tex] has an inverse that is also a function.
