Reciprocal & Rational functions (+ finding roots & asymptotes): notes and practice questions

Consider the function $g(x) = \frac{ax^2+x+k}{x-4}$.

The graph of $y=g(x)$ passes through the point $(1, -2)$ and has an oblique asymptote with equation $y = -3x-11$.

(a) Write down the equation of the vertical asymptote.

(b) Find the value of:

(i) $a$

(ii) $k$

(c) Hence, find the exact coordinates of any points where the graph of $y=g(x)$ intersects the x-axis.

In a study of wave propagation, a mathematical model uses the function $g(x) = \frac{e^x - e^{-x}}{2}$, where $x \in \mathbb{R}$, to describe a certain physical quantity. This function is also known as the hyperbolic sine function, $\sinh x$.

Verify that $y = g(x)$ satisfies the differential equation $\frac{d^2y}{dx^2} = y$.

Another related function, the hyperbolic cosine, is defined as $f(x) = \frac{e^x + e^{-x}}{2}$, also known as $\cosh x$. Show that $(\cosh x)^2 - (\sinh x)^2 = 1$.

The functions $\cosh x$ and $\sinh x$ can be extended to complex numbers. Using Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta \in \mathbb{R}$, express $\cosh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Similarly, express $\sinh(i\theta)$ in terms of $\cos \theta$ and $\sin \theta$.

Hence, show that $(\cosh(i\theta) )^2 + (\sinh(i\theta) )^2 = \cos(2\theta)$.

In a design project, a component's profile is described by a hyperbola with parametric equations $x = A \cosh t$ and $y = B \sinh t$, where $A, B$ are positive constants and $t \in \mathbb{R}$.

Given that the component's profile passes through the point $(6, 0)$ and has asymptotes $y = \pm \frac{4}{3}x$, find the values of $A$ and $B$.

Consider the function $h(x) = \frac{8-2x}{x+1}$.

(a) State the largest possible domain for the function $h$.

(b) State the largest possible range for the function $h$.

(c) Find the coordinates of any points where the curve $y = h(x)$ intersects the $x$ and $y$-axes.

(a) Show that $2x+5 + \frac{3}{x-1} = \frac{2x^2 + 3x - 2}{x-1}$, for $x \in \mathbb{R}, x \neq 1$.

(b) Hence or otherwise, solve the equation $2\sin{\theta} + 5 + \frac{3}{\sin{\theta}-1} = 0$ for $0 \leq \theta \leq 2\pi$, $\theta \neq \frac{\pi}{2}$.

A function $f$ is defined by $f(x) = \frac{4x-1}{2x+3}$, where $x \in \mathbb{R}, x \neq -\frac{3}{2}$.

The graph of $y = f(x)$ is shown below.

(a) Write down the equation of the horizontal asymptote.

Consider the function $g(x) = mx - \frac{1}{3}$, where $m \in \mathbb{R}, m \neq 0$.

(i) Write down the number of solutions to $f(x) = g(x)$ for $m < 0$.

(ii) Determine the value of $m$ such that $f(x) = g(x)$ has only one solution for $x$.

(iii) Determine the range of values for $m$ for which $f(x) = g(x)$ has two distinct solutions for $x \le 0$.

For each of the following functions, write down its greatest possible domain and range.

(a) $f(x) = \frac{5}{2x-8}$

(b) $g(x) = \frac{10x-1}{2x-8}$

(c) $h(x) = \frac{10x-1}{2x}$

(d) $p(x) = \frac{5}{2x}$

The function $g$ is defined by $g(x)=\frac{5x-10}{3x+6}$ for $x \in \mathbb{R}$, $x \neq -2$.

(a) Find the zero of $g(x)$.

(b) For the graph of $y = g(x)$, write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.

(c) Find $g^{-1}(x)$, the inverse function of $g(x)$.

The functions $f$ and $g$ are defined by

$f(x) = x+k$, where $k \in \mathbb{R}$

$g(x) = \frac{1}{x-1}$, where $x \neq 1$.

(a) State the equation of the vertical asymptote to the graph of $y = g(x)$.

The graphs of $y = f(x)$ and $y = g(x)$ intersect at two distinct points.

(i) Show that, at the points of intersection, $x^2 + (k-1)x - (k+1) = 0$.

(ii) Hence show that $k^2 + 2k + 5 > 0$.

(iii) Hence, or otherwise, find the range of possible values of $k$.

The following diagram shows part of the graphs of $y = f(x)$ and $y = g(x)$.

The graphs intersect at $x = p$ and $x = q$, where $p < q$.

In the case where $k=4$, find the value of $q - p$. Express your answer in the form $\sqrt{b}$, where $b \in \mathbb{Z}^+$.

Let $g(x) = \frac{3x+6}{x-2}$.

(a) Find the coordinates of the $x$-intercept.

(b) Write down the equation of the vertical asymptote.

(c) Find the equation of the horizontal asymptote.

Show that $3x + 2 - \frac{2}{x+1} = \frac{3x^2 + 5x}{x+1}$, for $x \in \mathbb{R}, x \neq -1$.

Hence or otherwise, solve the equation $3\cos(\theta) + 2 - \frac{2}{\cos(\theta)+1} = 0$ for $0 \le \theta \le 2\pi, \theta \neq \pi$.

A civil engineer is analyzing the structural integrity of a new bridge design. The deflection of a certain point on the bridge, $D(x)$, in millimeters, is modeled by the function $D(x) = \frac{2x+1}{x^2-4}$, where $x$ represents the horizontal distance in meters from a central support. The model is valid for $x \in \mathbb{R}$, $x\neq p$, $x\neq q$.

Find the value of $p$ and the value of $q$.

Find an expression for $D'(x)$.

The graph of $y = D(x)$ has exactly one point of inflexion.

Find the x-coordinate of the point of inflexion.

Sketch the graph of $y = D(x)$ for $-4 \leq x \leq 4$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima (if they exist), and giving the equations of any asymptotes.

Consider a related model for stress distribution, $S(x) = \frac{x^2-4}{2x+1}$ for $x \in \mathbb{R}$, $x \neq -\frac{1}{2}$.

Find the equations of all the asymptotes on the graph of $y = S(x)$.

The engineer needs to identify the regions where the bridge deflection $D(x)$ is less than $1$ mm. Solve $D(x) < 1$ for $x \in \mathbb{R}$.

A function is defined by $f(x) = \frac{ax+b}{x+d}$.

(a) The asymptotes of the graph of $y=f(x)$ are a vertical line and a horizontal line which intersect at the point $(-2, 4)$. Find the value of $a$ and the value of $d$.

(b) The graph of $y=f(x)$ passes through the point $(0, -3)$. Find the value of $b$.

The function $g$ is defined by $g(x)=\frac{5x-10}{3x+6}$ for $x \in \mathbb{R}, x \ne -2$.

(a) Find the zero of $g(x)$.

(b) For the graph of $y = g(x)$, write down the equation of

(i) the vertical asymptote;

(ii) the horizontal asymptote.

(c) Find $g^{-1}(x)$, the inverse function of $g(x)$.

A company models the efficiency of a new production line by the function $f(x) = \frac{2x+3}{x+2}$, where $f(x)$ represents the output rate (in units per hour) after $x$ hours of operation. For mathematical analysis, we consider the function over its natural domain $x \in \mathbb{R}$, $x \ne -2$.

For the graph of f,

write down the equation of the vertical asymptote;

find the equation of the horizontal asymptote.

Find $f^{-1}(x)$.

Using an algebraic approach, show that the graph of $f^{-1}$ is obtained by a reflection of the graph of f in the y-axis followed by a reflection in the x-axis.

The graphs of f and $f^{-1}$ intersect at $x = p$ and $x = q$, where $p < q$.

Find the value of p and the value of q.

Hence, find the area enclosed by the graph of f and the graph of $f^{-1}$.

A function $f$ is defined by $f(x) = \frac{a}{x-2} + k$, for $x \neq 2$.

(a) The graph of $y=f(x)$ has a horizontal asymptote with equation $y=1$. Write down the value of $k$.

(b) The graph of $y=f(x)$ has an $x$-intercept at $5$. Find the value of $a$.

(c) Find the coordinates of the $y$-intercept of the graph of $y=f(x)$.

Given that $x > 3$, show that $\frac{x^2 - 2x - 3}{x^2 - 1} \times \frac{x-1}{x+3} = \frac{x-3}{x+3}$.

Hence, or otherwise, solve $\log_3((x^2 - 2x - 3)(x-1) ) + 1 = \log_3((x^2 - 1)(x+3) )$.

A team of engineers is designing a new roller coaster ride. The path of a certain section of the ride can be modelled by the function $g(x)=\frac{x^2 + 3x - 10}{x-4}$, where $x$ is the horizontal distance in metres from the starting point and $g(x)$ is the vertical height in metres. The domain of the function is $x \in \mathbb{R}, x\neq 4$.

Find the coordinates where the path of the roller coaster crosses the horizontal ground (x-axis).

Find the coordinates where the path of the roller coaster crosses the vertical axis (y-axis).

Write down the equation of the vertical asymptote of the graph of $g$.

The oblique asymptote of the graph of $g$ can be written as $y = ax + b$ where $a, b \in \mathbb{Z}$.

Find the value of $a$ and the value of $b$.

Sketch the graph of $g$ for $-20 \le x \le 20$, clearly indicating the points of intersection with each axis and any asymptotes.

Engineers want to analyse the inverse of the roller coaster's height function, $h(x) = \frac{1}{g(x)}$.

Express $h(x)$ in partial fractions.

Hence find the exact value of $\int_{0}^{1} h(x) dx$, expressing your answer as a single logarithm.

Consider the function $f(x) = 5 - \frac{1}{x+b}$, where $b$ is a constant and $x \neq -b$.

(a) Write down the equation of the horizontal asymptote of the graph of $f$.

(b) The vertical and horizontal asymptotes of the graph of $f$ intersect at the point $(3, 5)$. Find the value of $b$.

The following diagram shows the graph of $y = f(x)$, where $f(x) = \frac{Cx + D}{x+2}$, for $x \in \mathbb{R}$, $x \neq -2$ and $C, D \in \mathbb{Z}$.

The graph has a y-intercept at $(0, -3)$ and an x-intercept at $(3, 0)$.

(a) Find the value of $C$ and the value of $D$.

(b) Describe a sequence of transformations that maps the graph of $g(x) = \frac{1}{x}$ onto the graph of $y = f(x)$.