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Math  /  Algebra

QuestionComplete the sentence below. If f−1f^{-1} denotes the inverse of a function ff, then the graphs of ff and f−1f^{-1} are symmetric with respect to the line \qquad .
If f−1f^{-1} denotes the inverse of a function ff, then the graphs of ff and f−1f^{-1} are symmetric with respect to the line □\square y=x2y=x+1y=xy=x2+1\begin{array}{l} y=x^{2} \\ y=x+1 \\ y=x \\ y=x^{2}+1 \end{array}

Studdy Solution

STEP 1

1. We are given that f−1 f^{-1} is the inverse of a function f f .
2. We need to determine the line of symmetry for the graphs of f f and f−1 f^{-1} .

STEP 2

1. Understand the property of inverse functions related to symmetry.
2. Identify the correct line of symmetry from the given options.

STEP 3

Recall that the graph of a function f f and its inverse f−1 f^{-1} are symmetric with respect to a specific line. This line is the line where each point (a,b)(a, b) on the graph of f f corresponds to a point (b,a)(b, a) on the graph of f−1 f^{-1} .

STEP 4

The line of symmetry for the graphs of a function and its inverse is the line y=x y = x . This is because the roles of x x and y y are swapped for the inverse function.
The sentence can be completed as follows:
If f−1 f^{-1} denotes the inverse of a function f f , then the graphs of f f and f−1 f^{-1} are symmetric with respect to the line y=x y = x .

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