Answer :
To determine which function has an inverse that is a function, we need to check if each original function is one-to-one. A function is one-to-one if it passes the Horizontal Line Test, meaning that any horizontal line intersects the graph of the function at most once.
Let's analyze each given function one by one:
1. b(x) = x^2 + 3
- This is a quadratic function. The graph of [tex]\(x^2 + 3\)[/tex] is a parabola that opens upwards.
- For quadratic functions, a horizontal line can intersect the parabola at two different points.
- Hence, [tex]\(b(x) = x^2 + 3\)[/tex] is not one-to-one and does not have an inverse that is a function.
2. d(x) = -9
- This is a constant function, [tex]\(d(x) = -9\)[/tex], meaning that the output value is -9 for any input value of x.
- A horizontal line intersecting at -9 will obviously intersect the function an infinite number of times.
- Therefore, [tex]\(d(x) = -9\)[/tex] is not one-to-one and does not have an inverse that is a function.
3. m(x) = -7x
- This is a linear function with a non-zero slope. The graph of [tex]\( -7x \)[/tex] is a straight line.
- For a linear function with a non-zero slope, any horizontal line will intersect the graph at exactly one point.
- Therefore, [tex]\(m(x) = -7x\)[/tex] is one-to-one and does have an inverse that is a function.
4. p(x) = |x|
- This is the absolute value function. The graph of [tex]\(|x|\)[/tex] forms a V shape.
- A horizontal line can intersect the V-shaped graph at two different points.
- Thus, [tex]\(p(x) = |x|\)[/tex], is not one-to-one and does not have an inverse that is a function.
From the analysis above, we see that the only function that has an inverse which is also a function is:
m(x) = -7x
Thus, the function that has an inverse that is a function corresponds to the choice number 3.
Let's analyze each given function one by one:
1. b(x) = x^2 + 3
- This is a quadratic function. The graph of [tex]\(x^2 + 3\)[/tex] is a parabola that opens upwards.
- For quadratic functions, a horizontal line can intersect the parabola at two different points.
- Hence, [tex]\(b(x) = x^2 + 3\)[/tex] is not one-to-one and does not have an inverse that is a function.
2. d(x) = -9
- This is a constant function, [tex]\(d(x) = -9\)[/tex], meaning that the output value is -9 for any input value of x.
- A horizontal line intersecting at -9 will obviously intersect the function an infinite number of times.
- Therefore, [tex]\(d(x) = -9\)[/tex] is not one-to-one and does not have an inverse that is a function.
3. m(x) = -7x
- This is a linear function with a non-zero slope. The graph of [tex]\( -7x \)[/tex] is a straight line.
- For a linear function with a non-zero slope, any horizontal line will intersect the graph at exactly one point.
- Therefore, [tex]\(m(x) = -7x\)[/tex] is one-to-one and does have an inverse that is a function.
4. p(x) = |x|
- This is the absolute value function. The graph of [tex]\(|x|\)[/tex] forms a V shape.
- A horizontal line can intersect the V-shaped graph at two different points.
- Thus, [tex]\(p(x) = |x|\)[/tex], is not one-to-one and does not have an inverse that is a function.
From the analysis above, we see that the only function that has an inverse which is also a function is:
m(x) = -7x
Thus, the function that has an inverse that is a function corresponds to the choice number 3.
