The question asks to find the explicit inverse function f⁻¹(x), evaluate f⁻¹ at a specific point, or use a given relationship involving the inverse to determine an unknown constant.
Let $f$ be the function defined above. $$f ( x ) = \begin{cases} \frac { ( 2 x + 1 ) ( x - 2 ) } { x - 2 } & \text { for } x \neq 2 \\ k & \text { for } x = 2 \end{cases}$$ For what value of $k$ is $f$ continuous at $x = 2$ ? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5
Many physiological and biochemical processes, such as heartbeats and respiration rate, present scales constructed from the relationship between surface area and mass (or volume) of the animal. One of these scales, for example, considers that ``the cube of the surface area $S$ of a mammal is proportional to the square of its mass $M$''. This is equivalent to saying that, for a constant $k > 0$, the area $S$ can be written as a function of $M$ by means of the expression: (A) $S = k \cdot M$ (B) $S = k \cdot M^{\frac{1}{3}}$ (C) $S = k^{\frac{1}{3}} \cdot M^{\frac{1}{3}}$ (D) $S = k^{\frac{1}{3}} \cdot M^{\frac{2}{3}}$ (E) $S = k^{\frac{1}{3}} \cdot M^{2}$
For a real number $k$, let $g ( x )$ be the inverse function of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x + k$. For the equation $4 f ^ { \prime } ( x ) + 12 x - 18 = \left( f ^ { \prime } \circ g \right) ( x )$ to have a real root in the closed interval $[ 0,1 ]$, let $m$ be the minimum value of $k$ and $M$ be the maximum value of $k$. Find the value of $m ^ { 2 } + M ^ { 2 }$. [4 points]
For the function $f ( x ) = \frac { k } { x - 3 } + 1$, when $f ^ { - 1 } ( 7 ) = 4$, what is the value of the constant $k$? (Note: $k \neq 0$) [3 points] (1) 2 (2) 4 (3) 6 (4) 8 (5) 10
9. The coordinates of the intersection point of the graph of the inverse function of $f ( x ) = \log _ { 3 } ( x + 3 )$ with the $y$-axis are $\_\_\_\_$.
8. For any positive number $a$ not equal to 1, the graph of the inverse function of $f ( x ) = \log _ { a } ( x + 3 )$ always passes through point $P$. Then the coordinates of point $P$ are $\_\_\_\_$ $(0, -2)$ Analysis: The graph of $f ( x ) = \log _ { a } ( x + 3 )$ passes through the fixed point $( - 2,0 )$, so the graph of its inverse function passes through the fixed point $( 0 , - 2 )$
(1) [4 marks] Given is the function $f$ defined on $\mathbb { R }$ with $f ( x ) = e ^ { 2 x + 1 }$. Show that $f$ is invertible and determine a term for the inverse function of $f$. (2a) [3 marks] Given is the function $g : x \mapsto \left( x ^ { 2 } - 9 x \right) \cdot \sqrt { 2 - x }$ with maximal domain $D _ { g }$. State $D _ { g }$ and all zeros of $g$. (2b) [3 marks] Given is the function $h : x \mapsto \ln \left( \frac { 1 } { x ^ { 2 } + 1 } \right)$ defined on $\mathbb { R }$. Justify that the range of $h$ is the interval ] $- \infty ; 0 ]$. Consider the function $f$ defined on $\mathbb { R } ^ { + }$ with $f ( x ) = \frac { 1 } { \sqrt { x ^ { 3 } } }$. (3a) [2 marks] Show that the function $F$ defined on $\mathbb { R } ^ { + }$ with $F ( x ) = - \frac { 2 } { \sqrt { x } }$ is an antiderivative of $f$. (3b) [3 marks] The graph of $f$ encloses an area with the x-axis and the lines with equations $x = 1$ and $x = b$ with $b > 1$. Determine the value of $b$ for which this area has content 1. Given are the function $f$ defined on $\mathbb { R }$ with $f ( x ) = \frac { 1 } { 8 } x ^ { 3 }$ and the points $Q _ { a } ( a \mid f ( a ) )$ for $a \in \mathbb { R }$. The figure shows the graph of $f$ as well as the points $P ( 0 \mid 2 )$ and $Q _ { 2 }$. [Figure] (4a) [2 marks] Calculate for $a \neq 0$ the slope $m _ { a }$ of the line through the points $P$ and $Q _ { a }$ as a function of $a$. (for verification: $m _ { a } = \frac { a ^ { 3 } - 16 } { 8 a }$ ) )$} The tangent to the graph of $f$ at the point $Q _ { a }$ is denoted by $t _ { a }$. Determine computationally the value of $a \in \mathbb { R }$ for which $t _ { a }$ passes through $P$. Given is the function $f : x \mapsto \frac { 6 x } { x ^ { 2 } - 4 }$ defined on $\mathbb { R } \backslash \{ - 2 ; 2 \}$. The graph of $f$ is denoted by $G _ { f }$ and is symmetric with respect to the origin.
Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Determine an equivalent of $W ( x )$ as $x \rightarrow 0$ as well as an equivalent of $W ( x )$ as $x \rightarrow + \infty$.
Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the equation with unknown $x \in \mathbb { R }$ $$x \mathrm { e } ^ { x } = m \tag{I.1}$$ Determine, as a function of $m$, the number of solutions of (I.1). Explicitly express the possible solutions using the functions $V$ and $W$.
108. The function $f(x) = x^2\sqrt{x}$ is one-to-one on a domain. Which of the following is the inverse function on this domain? (1) $-\sqrt{x^2}\ ,\ x \leq 0$ (2) $-\sqrt[3]{x}\ ,\ x \leq 0$ (3) $-\sqrt{x^2}\ ,\ x \geq 0$ (4) $-\sqrt[3]{x}\ ,\ x \geq 0$
28. If $f ( x ) = 3 x - 5$, then $f - 1 ( x )$ (A) is given by $1 / ( 3 x - 5 )$. (B) is given by $( x + 5 ) / 3 \quad$. (C) does not exist because $f$ is not one-one (D) does not exist because $f$ is not onto.
The number of solutions of the equation $\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right) = \sin^{-1}\left(\frac{x-1}{\sqrt{1+(x-1)^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+(x-1)^2}}\right)$ is (A) 0 (B) 1 (C) 2 (D) infinite
The function $f : R \sim \{ 0 \} \rightarrow R$ given by $f ( x ) = \frac { 1 } { x } - \frac { 2 } { e ^ { 2 x } - 1 }$ can be made continuous at $x = 0$ by defining $f ( 0 )$ as (1) 2 (2) - 1 (3) 0 (4) 1
If $R = \{(x, y) : x, y \in Z, x^{2} + 3y^{2} \leq 8\}$ is a relation on the set of integers $Z$, then the domain of $R^{-1}$ is (1) $\{-2, -1, 1, 2\}$ (2) $\{0, 1\}$ (3) $\{-2, -1, 0, 1, 2\}$ (4) $\{-1, 0, 1\}$