Math Problem Statement

Use the graph of f to draw the graph of its inverse function. . . . Question content area top right Part 1 -6 6 -6 6

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A curve passes through the points (negative 6, negative 2.7), (0, 0), and (1, 1.5). The curve passes through the plotted points left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 3.8 right parenthesis . All coordinates are approximate. Question content area bottom Part 1 Choose the correct graph of the inverse function f Superscript negative 1 below. The graph of f is dashed in each of the choices. A. -6 6 -6 6

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A dashed curve passes through the points (negative 6, negative 2.7), (0, 0), and (1, 1.5). The curve passes through the plotted points left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 3.8 right parenthesis . A solid curve passes through the points (negative 2.7, negative 6), (0, 0), and (1.5, 1). The curve passes through the plotted points left parenthesis 0 comma 0 right parenthesis and left parenthesis 3.8 comma 2 right parenthesis . All coordinates are approximate. B. -6 6 -6 6

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A dashed curve passes through the points (negative 6, negative 2.7), (0, 0), and (1, 1.5). The curve passes through the plotted points left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 3.8 right parenthesis . A solid curve passes through the points (2.7, 6), (0, 0), and (negative 1.5, negative 1). The curve passes through the plotted points left parenthesis 0 comma 0 right parenthesis and left parenthesis negative 3.8 comma negative 2 right parenthesis . All coordinates are approximate. C. -6 6 -6 6

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. A dashed curve passes through the points (negative 6, negative 2.7), (0, 0), and (1, 1.5). The curve passes through the plotted points left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 3.8 right parenthesis . A solid curve passes through the points (negative 2.7, 6), (0, 0), and (1.5, negative 1). The curve passes through the plotted points left parenthesis 0 comma 0 right parenthesis and left parenthesis 3.8 comma negative 2 right parenthesis . All coordinates are approximate. D. -6 6 -6 6

Solution

To determine the correct graph for the inverse function f−1f^{-1}, we need to apply the principle that the graph of an inverse function is a reflection of the original function ff across the line y=xy = x. This means that if the original function ff passes through points like (a,b)(a, b), then its inverse f−1f^{-1} should pass through points (b,a)(b, a).

Given that the function ff passes through the approximate points:

  1. (−6,−2.7)(-6, -2.7)
  2. (0,0)(0, 0)
  3. (1,1.5)(1, 1.5)
  4. (2,3.8)(2, 3.8)

For the inverse f−1f^{-1}, the points will be:

  1. (−2.7,−6)(-2.7, -6)
  2. (0,0)(0, 0)
  3. (1.5,1)(1.5, 1)
  4. (3.8,2)(3.8, 2)

Let's match these points with the options:

  1. Option A: The solid curve passes through (−2.7,−6)(-2.7, -6), (0,0)(0, 0), (1.5,1)(1.5, 1), and (3.8,2)(3.8, 2), which are correct points for f−1f^{-1}.
  2. Option B: The solid curve passes through points like (2.7,6)(2.7, 6), which does not match the required points.
  3. Option C: The solid curve passes through points like (−2.7,6)(-2.7, 6) and (1.5,−1)(1.5, -1), which do not match the required points.
  4. Option D: This option is missing, so we do not have enough data to evaluate it.

Conclusion: The correct answer is Option A, as it accurately reflects the points needed for the inverse function f−1f^{-1} based on the given points on ff.

Would you like more details or have any questions on this process?

Here are some additional questions related to inverse functions:

  1. What are the properties of an inverse function in relation to the original function?
  2. How do you verify if two functions are inverses of each other?
  3. Why is the line y=xy = x the line of reflection for a function and its inverse?
  4. How does one determine if a function has an inverse by looking at its graph?
  5. Can every function have an inverse? If not, why?

Tip: When sketching the graph of an inverse function, remember to switch the x- and y-coordinates of each point on the original function.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Reflection Across y=x

Formulas

Inverse of a function: If f(x) passes through (a, b), then f^{-1}(x) passes through (b, a)

Theorems

Reflection property of inverse functions across y = x

Suitable Grade Level

Grades 10-12

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