Solving equations [f(x)=0 or f(x)=g(x)] (graphically, analytically, GDC): notes and practice questions

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 $y=f(x)$.

Consider the function $h(x) = e^{2x} - e^{-2x}$, for $x \in \mathbb{R}$.

Show that $h$ is an odd function.

The function $k$ is given by $k(x) = \frac{x+1}{x^2 - 2x - 8}$, where $x \in \mathbb{R}, x \neq -2, x \neq 4$.

Solve the inequality $h(x) \ge k(x)$.

Consider the function $h(x) = \ln\left(\frac{1+x}{1-x}\right)$, for $x \in (-1, 1)$.

(a) Show that $h$ is an odd function.

The function $k$ is given by $k(x) = \frac{3x}{x^2 - 0.25}$, where $x \in \mathbb{R}, x \neq \pm 0.5$.

(b) Solve the inequality $h(x) \ge k(x)$.

Consider the functions $f(x) = \ln(x-p)$ and $g(x) = \frac{1}{4}x^2 + q$ where $p, q \in \mathbb{R}$.

(a) Find $g'(x)$.

The graphs of $f$ and $g$ have a common tangent at the point where $x = 2$.

(b) Show that $p = 1$.

(c) Hence, find the value of $q$.

A biologist is modelling the growth of two different bacterial colonies. The first colony, A, grows such that its population at time $x$ is given by $P_A(x) = a^x$, where $a$ is a growth factor and $x \ge 0$. The second colony, B, grows linearly such that its population at time $x$ is $P_B(x) = x$.

Consider the cases where the growth factor $a = 2$ and $a = 10$. On the same set of axes, sketch the following three graphs for $x \ge 0$:

$y = 2^x$

$y = 10^x$

$y = x$

Clearly label each graph with its equation and state the coordinates of any non-zero $y$-axis intercepts.

In parts (b) and (c), consider the case where the growth factor $a = e$.

Use calculus to find the minimum value of the expression $e^x - x$, justifying that this value is a minimum.

Hence deduce that $e^x > x$ for all $x \in \mathbb{R}$.

There exist values of $a$ for which the graph of $y = a^x$ and the line $y = x$ have different numbers of intersection points. The following table gives three intervals for the value of $a$.

By investigating the graph of $y = a^x$ for different values of $a$, write down the values of $p, q$ and $r$.

In parts (e) and (f), consider $a \in \mathbb{R}^+, a \neq 1$.

For $1.4 \leq a \leq 1.5$, a value of $a$ exists such that the line $y = x$ is a tangent to the graph of $y = a^x$ at a point P.

Find the exact coordinates of P and the exact value of $a$.

Write down the exact set of values for $a$ such that the graphs of $y = a^x$ and $y = x$ have

(i) two intersection points;

(ii) no intersection points.

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 $g(x)$.

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 $x \le 0$.

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

$f(x) = 2\ln x$, where $x > 0$

$g(x) = \ln(k(x-2) )$, where $x > 2, k \in \mathbb{R}^+$.

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

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

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

(b) (ii) Hence show that $k^2 - 8k > 0$.

(b) (iii) Find the range of possible values of $k$.

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

(c) In the case where $k=10$, find the value of $q-p$. Express your answer in the form $a\sqrt{b}$, where $a, b \in \mathbb{Z}^+$.

A function is defined by $f(x) = \frac{1}{2}x^2 + x + 4$. The following diagram shows part of the graph of $f$.

The graph has a vertex at V and intersects the y-axis at point P.

(a) Find the coordinates of the vertex V.

(b) Write down the coordinates of the y-intercept, P.

(c) The line L is the normal to the graph of $f$ at point P. Find the equation of L, giving your answer in the form $y=mx+c$.

(d) The line L intersects the graph of $f$ at a second point, Q. Calculate the distance between P and Q.

Consider the function $f$ defined by $f(x) = x^3 - 6x^2 + 8x$.

Find the $x$-intercepts of the graph of $y=f(x)$.

The graph of $y=f(x)$ for $0 \le x \le 4$ is shown below. The graph encloses two regions with the $x$-axis, shaded in the diagram.

Find the total area of the shaded regions.

The total surface area of a closed right cylinder is 8, equal to the total shaded area found in part (b). The cylinder has a height of $\frac{4-\pi}{\pi}$.

Find the radius, $r$, of the cylinder.

Hence, find the volume of the cylinder.

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 $b \in \mathbb{Z}^+$.

The functions $f$ and $g$ are defined by $f(x) = \sin x$ and $g(x) = \cot x$, for $0 < x < \frac{\pi}{2}$.

The curves $y = f(x)$ and $y = g(x)$ intersect at a point P whose x-coordinate is $k$.

Show that $\sin^2 k = \cos k$.

Hence, show that the tangent to the curve $y = f(x)$ at P and the tangent to the curve $y = g(x)$ at P are perpendicular.

Find the value of $\cos k$. Give your answer in the form $\frac{a+\sqrt{b}}{c}$, where $a, c \in \mathbb{Z}$ and $b \in \mathbb{Z}^+$.

The graph of the function $f(x) = x^3 - 5x^2 + 8x$ and the line $L$ with equation $y=4x$ are shown in the diagram below. The graphs intersect at the origin O, and at points A and B.

(a) Find the coordinates of A and B.

(b) The region enclosed by the graph of $f(x)$ and the line $L$ is composed of two smaller regions. Find the total area of these two enclosed regions.

Consider the functions $f(x) = k - (x-h)^2$ and $g(x) = \ln(x-1) + 2$ where $h, k \in \mathbb{R}$.

The graphs of $f$ and $g$ have a common tangent at $x=2$.

(a) Find $g'(x)$.

(b) Show that $h = \frac{5}{2}$.

(c) Hence, find the value of $k$.

Consider the function f defined by $f(x) = 50e^{-0.2x}$ for $x \in \mathbb{R}^+$.

The graph of f and the line $y = x$ intersect at point P.

Find the x-coordinate of P.

The line L has a gradient of -2 and is a tangent to the graph of f at the point Q.

Find the exact coordinates of Q.

Show that the equation of L is $y = -2x + 10 \ln 5 + 10$.

The shaded region A is enclosed by the graph of f and the lines $y = x$ and L.

Find the x-coordinate of the point where L intersects the line $y = x$.

Hence, find the area of A.

The line L is tangent to the graphs of both f and the inverse function $f^{-1}$.

Find the shaded area enclosed by the graphs of f and $f^{-1}$ and the line L.

Consider the function f defined by $f(x) = \ln(x^2 - 3)$ for $x > \sqrt{3}$.

The following diagram shows part of the graph of f which crosses the x-axis at point A, with coordinates $(p, 0)$. The line L is the tangent to the graph of f at the point B.

(a) Find the exact value of $p$.

(b) Given that the gradient of L is $1$, find the x-coordinate of B.

The rate of change of a certain quantity $R$ with respect to a variable $x$ is given by $R'(x)=\frac{1}{x(M-x)}$, $x \in \mathbb{R}$, $x \neq 0$, $x \neq M$ where $M$ is a positive constant.

The expression for $R'(x)$ can be written in the form $\frac{A}{x} + \frac{B}{M-x}$, where $A, B \in \mathbb{R}$.

Find $A$ and $B$ in terms of $M$.

Hence, find an expression for $R(x)$.

The concentration of a certain chemical product, $C$ (in mol/L), in a reaction vessel at time $t$ (in minutes) can be modelled by the differential equation $\frac{dC}{dt} = \frac{C(L-C)}{8L}$, where $L$ is the maximum possible concentration and $C(0) = 0.2$ mol/L is the initial concentration.

By solving the differential equation, show that $C = \frac{0.2 L}{(L-0.2)e^{-\frac{t}{8}}+0.2}$.

At $t=12$ minutes, the concentration of the product has reached $0.6$ mol/L.

Find the value of $L$, giving your answer correct to four significant figures.

Find the value of $t$ when the rate of change of the concentration is at its maximum.

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 $g(x)$.

All answers in this question should be given to four significant figures.

A popular online game offers players a 'Mystery Box' for £5. Each box contains a prize, with the probability distribution for the prize value $D$ shown in the following table. For example, the probability of a player receiving £$25$ is 0.04. The initial grand prize in the first week of the game is £$500$.

(a) Find the value of $c$.

(b) Determine whether purchasing a mystery box in the first week is a fair game. Justify your answer.

(c) If the grand prize is not won and continues to triple each week, while all other prize amounts and probabilities remain the same, write an expression in terms of $n$ for the value of the grand prize in the $n$th week of the game.

(d) The $w$th week is the first week in which a player is expected to make a profit from purchasing a mystery box. If a player purchases a mystery box in the $w$th week, their expected profit is $p$.

Find the value of $p$.

A function $f$ is defined by $f(x) = \frac{\ln(x)}{x}$, for $x > 0$.

The following diagram shows part of the graph of $f$.

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

(b) Find $f'(x)$.

The graph of $f$ has a local maximum at point M.

(c) Hence, find the exact coordinates of M.

(d) (i) Show that $f''(x) = \frac{2\ln(x) - 3}{x^3}$.

The graph of $f$ has a point of inflection at point P.

(d) (ii) Hence, find the exact coordinates of P.