Inverse functions

Question 1: Correct Statements about Inverse Functions (from file image_c61b4b.png)

The Question: Which of the following options is(are) CORRECT for a reversible function $f$?

Detailed Solution:

Let’s evaluate each statement based on the core properties.

• “For any function f, $f^{-1}(x) = \frac{1}{f(x)}$”: FALSE. This is a common notational confusion. $f^{-1}$ denotes the inverse operation, not the multiplicative reciprocal.
• “The domain of a function $f$ is always equal to the range of $f^{-1}$ function”: TRUE. This is a fundamental property of inverse functions; the inputs of the original are the outputs of the inverse.
• “For any x = a in the domain of a function f, if (a, f(a)) is on the graph of f, then (f(a), a) is on the graph of $f^{-1}$”: TRUE. This describes exactly how the coordinates of points are swapped between a function and its inverse.
• “For any function f, $(f \circ f^{-1})(x) = x$”: TRUE. This is the composition property, showing that the functions “undo” each other.

Final Answer: The three correct statements are:

• The domain of a function $f$ is always equal to the range of $f^{-1}$ function
• For any x = a in the domain of a function f, if (a, f(a)) is on the graph of f, then (f(a), a) is on the graph of $f^{-1}$
• For any function f, $(f \circ f^{-1})(x) = x$

Question 2: Domain and Range of an Inverse Function (from file image_c61b4b.png)

The Question: If $f(x) = \sqrt{x-5}$, then the domain and the range of the function $f^{-1}$ are respectively __________.

Detailed Solution:

We can use the domain/range swapping property.

1. Find the domain of the original function, $f(x)$:

   • The expression inside the square root must be non-negative.
   • $x - 5 \ge 0 \implies x \ge 5$.
   • The domain of $f(x)$ is $[5, \infty)$.

2. Find the range of the original function, $f(x)$:

   • The output of the principal square root function is always non-negative.
   • The smallest possible output is $\sqrt{5-5} = 0$.
   • The range of $f(x)$ is $[0, \infty)$.

3. Determine the domain and range of the inverse, $f^{-1}(x)$:

   • Domain of $f^{-1}$ = Range of $f = [0, \infty)$.
   • Range of $f^{-1}$ = Domain of $f = [5, \infty)$.

Final Answer: The domain of $f^{-1}$ is $[0, \infty)$ and the range is $[5, \infty)$.

Question 3: Functions that are their Own Inverse (from file image_c61b4b.png)

The Question: Which of the following functions satisfy the condition $f(x) = f^{-1}(x)$?

Detailed Solution: We will find the inverse for each function and see if it matches the original.

• $f(x) = \frac{1}{x^3}$:

  • $y = \frac{1}{x^3} \implies$ Swap: $x = \frac{1}{y^3} \implies y^3 = \frac{1}{x} \implies y = \sqrt[3]{\frac{1}{x}}$. Not the same.

• $f(x) = \frac{1}{x}$:

  • $y = \frac{1}{x} \implies$ Swap: $x = \frac{1}{y} \implies y = \frac{1}{x}$. This is its own inverse.

• $f(x) = \frac{2x-1}{3x-2}$:

  • $y = \frac{2x-1}{3x-2} \implies$ Swap: $x = \frac{2y-1}{3y-2} \implies x(3y-2) = 2y-1 \implies 3xy - 2x = 2y-1 \implies 3xy - 2y = 2x - 1 \implies y(3x-2) = 2x-1 \implies y = \frac{2x-1}{3x-2}$. This is its own inverse.

• $f(x) = 9 - 5x$:

  • $y = 9 - 5x \implies$ Swap: $x = 9 - 5y \implies 5y = 9 - x \implies y = \frac{9-x}{5}$. Not the same.

• $f(x) = -x$:

  • $y = -x \implies$ Swap: $x = -y \implies y = -x$. This is its own inverse.

Final Answer: The functions that are their own inverses are:

• $f(x) = \frac{1}{x}$
• $f(x) = \frac{2x-1}{3x-2}$
• $f(x) = -x$

Question 4: Incorrect Statements about Inverses (from file image_c61af5.png)

The Question: Which of the following statements is INCORRECT?

Detailed Solution: We are looking for the statements that are false.

• “The inverse function(g) of a function $f(x) = \frac{1}{x^3}$ is $g(x) = \frac{1}{\sqrt{x}}$”:

  • Let’s find the inverse of $f(x) = 1/x^3$.
  • $y = 1/x^3 \implies$ Swap: $x = 1/y^3 \implies y^3=1/x \implies y = \sqrt[3]{1/x} = 1/\sqrt[3]{x}$.
  • The correct inverse is $1/\sqrt[3]{x}$, not $1/\sqrt{x}$. This statement is FALSE.

• “If $f(x) = \frac{1}{x-11}$ and $g(x) = 11 - \frac{1}{x}$, then $f(x)=g^{-1}(x)$ for all $x \in \mathbb{R}$”:

  • This asks if $f$ is the inverse of $g$. Let’s find the inverse of $g(x)$.
  • $y = 11 - \frac{1}{x} \implies$ Swap: $x = 11 - \frac{1}{y} \implies \frac{1}{y} = 11 - x \implies y = \frac{1}{11-x}$.
  • The inverse $g^{-1}(x) = \frac{1}{11-x}$, which is not the same as $f(x)=\frac{1}{x-11}$. This statement is FALSE.

• "$(f \circ f^{-1})(x) = x = (f^{-1} \circ f)(x)$, for all $x$ in the domain of the function $f$.": This is the correct statement of the composition property of inverse functions. This statement is TRUE.

• “A function $f$ is symmetric to function $f^{-1}$ about the line $y=x$.”: This is the correct statement of the graphical property of inverse functions. This statement is TRUE.

Final Answer: The incorrect statements are:

• The inverse function(g) of a function $f(x) = \frac{1}{x^3}$ is $g(x) = \frac{1}{\sqrt{x}}$
• If $f(x) = \frac{1}{x-11}$ and $g(x) = 11 - \frac{1}{x}$, then $f(x)=g^{-1}(x)$ for all $x \in \mathbb{R}$

Question 5: Identifying Inverse Pairs (from file image_c61af5.png)

The Question: If $f(x)=x^3-5, g(x)=\sqrt{x}+5, h(x)=(x-5)^3$ and $k(x)=\sqrt[3]{x+5}$ are functions, then which of the following options is(are) true?

Detailed Solution: We need to find the inverse for one function in each pair and see if it matches the other.

• Check k(x) = f^{-1}(x):

  • Find the inverse of $f(x) = x^3 - 5$.
  • $y = x^3 - 5$
  • Swap variables: $x = y^3 - 5$
  • Solve for y: $x+5 = y^3 \implies y = \sqrt[3]{x+5}$.
  • This result, $\sqrt[3]{x+5}$, is exactly the definition of $k(x)$. This statement is TRUE.

• Check f(x) = h^{-1}(x):

  • Find the inverse of $h(x) = (x-5)^3$.
  • $y = (x-5)^3$
  • Swap variables: $x = (y-5)^3$
  • Solve for y: $\sqrt[3]{x} = y-5 \implies y = \sqrt[3]{x} + 5$.
  • This is not equal to $f(x) = x^3 - 5$. This statement is FALSE.

• Check g(x) = k^{-1}(x):

  • Find the inverse of $k(x) = \sqrt[3]{x+5}$.
  • $y = \sqrt[3]{x+5}$
  • Swap variables: $x = \sqrt[3]{y+5}$
  • Solve for y: $x^3 = y+5 \implies y = x^3 - 5$.
  • This is equal to $f(x)$, not $g(x)$. This statement is FALSE.

• Check h(x) = g^{-1}(x):

  • Find the inverse of $g(x) = \sqrt{x} + 5$.
  • $y = \sqrt{x} + 5$
  • Swap variables: $x = \sqrt{y} + 5$
  • Solve for y: $x - 5 = \sqrt{y} \implies y = (x-5)^2$.
  • This is not equal to $h(x) = (x-5)^3$. This statement is FALSE.

Final Answer: The only true option is $k(x) = f^{-1}(x)$, for all $x \in \mathbb{R}$.