Math Problem Statement

Use the graph of f to draw the graph of its inverse function.

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Part 1

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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, 0.7), (0, negative 2), and (1, negative 3.5). The curve passes through the plotted points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 2 comma negative 5.8 right parenthesis . All coordinates are approximate.

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Part 1

Choose the correct graph of the inverse function

f Superscript negative 1f−1

below. The graph of f is dashed in each of the choices.

A.

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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, 0.7), (0, negative 2), and (1, negative 3.5). The curve passes through the plotted points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 2 comma negative 5.8 right parenthesis . A solid curve passes through the points (negative 0.7, 6), (2, 0), and (3.5, negative 1). The curve passes through the plotted points left parenthesis 2 comma 0 right parenthesis and left parenthesis 5.8 comma negative 2 right parenthesis . All coordinates are approximate.

B.

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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, 0.7), (0, negative 2), and (1, negative 3.5). The curve passes through the plotted points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 2 comma negative 5.8 right parenthesis . A solid curve passes through the points (negative 0.7, negative 6), (2, 0), and (3.5, 1). The curve passes through the plotted points left parenthesis 2 comma 0 right parenthesis and left parenthesis 5.8 comma 2 right parenthesis . All coordinates are approximate.

C.

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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, 0.7), (0, negative 2), and (1, negative 3.5). The curve passes through the plotted points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 2 comma negative 5.8 right parenthesis . A solid curve passes through the points (0.7, 6), (negative 2, 0), and (negative 3.5, negative 1). The curve passes through the plotted points left parenthesis negative 2 comma 0 right parenthesis and left parenthesis negative 5.8 comma negative 2 right parenthesis . All coordinates are approximate.

D.

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Solution

To find the correct graph of the inverse function, f−1f^{-1}, we need to understand that the inverse function reflects the original function ff over the line y=xy = x. This reflection means that for any point (a,b)(a, b) on the graph of ff, the corresponding point (b,a)(b, a) will appear on the graph of f−1f^{-1}.

Given Points on ff:

The points on the function ff are approximately:

  • (−6,0.7)(-6, 0.7)
  • (0,−2)(0, -2)
  • (1,−3.5)(1, -3.5)
  • (2,−5.8)(2, -5.8)

Reflected Points for f−1f^{-1}:

Reflect each of these points to get points for the inverse function f−1f^{-1}:

  • (−6,0.7)(-6, 0.7) becomes (0.7,−6)(0.7, -6)
  • (0,−2)(0, -2) becomes (−2,0)(-2, 0)
  • (1,−3.5)(1, -3.5) becomes (−3.5,1)(-3.5, 1)
  • (2,−5.8)(2, -5.8) becomes (−5.8,2)(-5.8, 2)

Analyzing Options

Using these transformed points:

  1. Option A has reflected points such as (0.7,6)(0.7, 6), which does not match the point (0.7,−6)(0.7, -6) expected for the inverse function.
  2. Option B has a point (0.7,−6)(0.7, -6), (2,0)(2, 0), and (3.5,1)(3.5, 1), which do not align with the reflection.
  3. Option C has points that align correctly with the inverse reflections:
    • (0.7,6)(0.7, 6) reflects correctly with the original point structure.

After analyzing the given options

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Math Problem Analysis

Mathematical Concepts

Graphing
Inverse Functions
Reflection Over Line y = x

Formulas

Reflection property: If (a, b) is on f, then (b, a) is on f^-1

Theorems

Inverse Function Theorem
Reflection Across y = x

Suitable Grade Level

Grades 9-12

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