Math

Question When can a quadratic function have an inverse? A) When its range is restricted B) When its domain is restricted C) When all xx-values are >1>-1 D) When all yy-values are >1>-1

Studdy Solution

STEP 1

Assumptions
1. A quadratic function is a function of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c where aa, bb, and cc are constants and a0a \neq 0.
2. An inverse function, denoted as f1(x)f^{-1}(x), exists if and only if the original function is one-to-one (bijective), meaning every yy-value has exactly one corresponding xx-value.
3. A function's domain is the set of all possible xx-values, and its range is the set of all possible yy-values that the function can output.

STEP 2

Understand the concept of a one-to-one function.
A function is one-to-one if it passes the horizontal line test, which means that no horizontal line intersects the graph of the function at more than one point.

STEP 3

Recognize the shape of a quadratic function's graph.
The graph of a quadratic function is a parabola. If a>0a > 0, the parabola opens upwards, and if a<0a < 0, it opens downwards.

STEP 4

Analyze the one-to-one nature of a quadratic function.
Because a parabola is symmetric with respect to its axis of symmetry (the line that passes through the vertex), it will not pass the horizontal line test for its entire domain. This means that a quadratic function is not one-to-one over its entire domain.

STEP 5

Consider the effect of restricting the domain.
By restricting the domain of a quadratic function to one side of the vertex (either left or right), we can make the function one-to-one because it will then pass the horizontal line test.

STEP 6

Consider the effect of restricting the range.
Restricting the range of a quadratic function does not ensure that the function becomes one-to-one. Even if the range is restricted, the function can still fail the horizontal line test.

STEP 7

Evaluate the given options in the context of a quadratic function having an inverse.
A. When its range is restricted - This option is incorrect because restricting the range does not guarantee a one-to-one function. B. When its domain is restricted - This option is correct because restricting the domain to one side of the vertex can make the function one-to-one. C. When all the xx-values are greater than -1 - This option is incorrect because the specific value of xx does not determine if the function is one-to-one. D. When all the yy-values are greater than -1 - This option is incorrect because the specific value of yy does not determine if the function is one-to-one.

STEP 8

Select the correct answer.
The correct answer is B: When its domain is restricted.
A quadratic function can have an inverse when its domain is restricted to ensure that it is a one-to-one function.

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