Consider the graph of g\left( x \right):
Find g \left( - 4 \right).
Find g \left( 2 \right).
Find g \left( - 6 \right).
Find x when g \left( x \right) = - 1.
For each of the following functions:
Sketch the graph of the function.
State the domain of the function.
State the range of the function.
f(x) = \begin{cases} -x, & x \geq 0 \\ x, & x \lt 0 \end{cases}
f(x) = \begin{cases} x - 3, & x \geq 3 \\ 3 - x, & x \lt 3 \end{cases}
f(x) = \begin{cases} 2 + x, & x \geq 0 \\ 2 - x, & x \lt 0 \end{cases}
f(x) = \begin{cases} x, & x \geq 1 \\ 2 - x, & x \lt 1 \end{cases}
f(x) = \begin{cases} \dfrac{2}{3}x + 4, & x \lt 0 \\ x + 4, & 0 \leq x \leq 1 \\ -2x + 7, & x \gt 1 \end{cases}
f(x) = \begin{cases} -x - 3, & x \lt -1 \\ x - 1, & -1 \leq x \leq 3 \\ 3x - 7, & x \gt 3 \end{cases}
f(x) = \begin{cases} x^{2} + 2, & x \geq 0 \\ 2 - x, & x \lt 0 \end{cases}
f(x) = \begin{cases} x + 2, & x \lt -2 \\ x^{2} - 4, & -2 \leq x \leq 2 \\ x - 2, & x \gt 2 \end{cases}
f(x) = \begin{cases} x^{2}, & x \lt 1 \\ -x, & x \geq 1 \end{cases}
f(x) = \begin{cases} 0, & x \lt -1 \\ -x^{3}, & x \geq -1 \end{cases}
f(x) = \begin{cases} 4x, & x \lt -1 \\ - 4, & -1 \leq x \leq 1 \\ -4x, & x \gt 1 \end{cases}
Sketch the graph of the following functions:
f(x) = \begin{cases} 1, & x \gt 1 \\ -3 + x, & x \leq 1 \end{cases}
f(x) = \begin{cases} 2x, & -4 \leq x \leq -1 \\ x^{2} + 1, & -1 \lt x \leq 2 \\ 5, & 2 \lt x \leq 8 \end{cases}
f(x) = \begin{cases} x^{2} - 8, & -3 \leq x \leq -1 \\ x^{3} - 5, & -1 \lt x \lt 2 \\ x - 9, & 2 \leq x \leq 4 \end{cases}
Consider the graph of f \left( x \right):
State the domain of f \left( x \right).
State the range of f \left( x \right).
Evaluate f \left( - 3 \right).
What are the two points where the graph of f \left( x \right) crosses the x-axis?
At what point does the graph of f \left( x \right) cross the y-axis?
For what values of x is f \left( x \right) < 0?
Consider the function:
f(x) = \begin{cases} 2x + 2, & x \lt 0 \\ 2, & 0 \leq x \leq 1 \\ -2x + 5, & x \gt 1 \end{cases}
State the domain of the function.
State the y-coordinate of the y-intercept of this function.
Sketch the graph of the function.
State the range of the function.
Is the function f continuous on its domain?
Define the function represented by the each of the following graphs:
Consider the function given, and solve for the value of k that will make the function continuous for all values in its domain.
f(x) = \begin{cases} 2x + k, & x \lt 1 \\ \dfrac{6}{x}, & 1 \leq x \leq 3 \end{cases}
The following function is continuous for all values in its domain:
f(x) = \begin{cases} m, & x \lt -2 \\ x + 4, & -2 \leq x \leq 2 \\ n, & x \gt -2 \end{cases}
Solve for m.
Solve for n.
Sketch the graph of the function.
For each of the following graphs:
Define the function represented by the graph.
State the domain of the function.
State the range of the function.
Consider the functions f \left( x \right) = 2 x and g \left( x \right) = 4.
If y is defined as y = f \left( x \right) + g \left( x \right), state the equation for y.
Sketch the graph of the function y.
Describe the transformation of f \left( x \right) that corresponds to y.
The functions f \left( x \right) and g \left( x \right) have been graphed as shown:
A function y is defined as y = f \left( x \right) - g \left( x \right).
Complete the given table of the corresponding points on the function defined by \\y = f \left( x \right) - g \left( x \right):
| x | -2 | -1 | 0 | -1 | -2 |
|---|---|---|---|---|---|
| f \left( x \right) - g \left( x \right) |
Hence, sketch the graph of the function y.
Sketch the graph of f(x)+g(x) for each of the following:
f(x)=4 - x and g(x)=x^2 -3
f \left( x \right) = 2 x^{2} and g \left( x \right) = 4 x + 6
f \left( x \right) = \dfrac{1}{x} and g \left( x \right) = 2 x
f \left( x \right) = \sqrt{x} and g \left( x \right) = 2 x + 2
For each of the following graphs, sketch y=f(x) + g(x) on a number plane:
Consider the functions f \left( x \right) = 4 \sqrt{x} and g \left( x \right) = x:
Sketch the graph of their sum: f \left( x \right) + g \left( x \right).
Sketch the graph of their difference: f \left( x \right) - g \left( x \right).
Sketch the graph of f(x)-g(x) for each of the following:
f \left( x \right) = x^{2} and g \left( x \right) = 2 x^{3}
f \left( x \right) = \dfrac{1}{x^{2}} and g \left( x \right) = 3 x
f \left( x \right) = x^{3} and g \left( x \right) = \dfrac{1}{x + 2}
For each of the following graphs, sketch y=f(x) - g(x) on a number plane:
Sketch the graph of the product of the following pairs of functions:
f \left( x \right) = x and g \left( x \right) = 2 x^{2} + 4
f \left( x \right) = 3 x and g \left( x \right) = 2 x^{2}
f \left( x \right) = \dfrac{1}{x} and g \left( x \right) = 6 x - 2
f \left( x \right) = \dfrac{1}{x^{3}} and g \left( x \right) = x^{3} - 3
For each of the following graphs, sketch y=f(x) \times g(x) on a number plane:
Consider the functions f \left( x \right) = x^{2} and g \left( x \right) = x + 5.
If y is defined as y = f \left( g \left( x \right) \right), state the equation for y.
Sketch the graph of the function y.
Describe the transformation of f \left( x \right) that corresponds to y.
Consider the functions f \left( x \right) = 4^{x} and g \left( x \right) = x - 3.
If y is defined as y = f \left( g \left( x \right) \right), state the equation for y.
The functions f \left( x \right) and y have been graphed on the same set of axes:
Describe the transformation of f \left( x \right) that corresponds to y.
Evaluate f \left( g \left( 2 \right) \right), given the following functions:
\begin{aligned} f \left( x \right) &= 4 x - 10 \\ g \left( x \right) &= 3 + \dfrac{3}{x}\end{aligned}Consider the following functions:
\begin{aligned} f \left( x \right) = - 2 x - 6 \\ g \left( x \right) = 5 x - 7 \end{aligned}Find f \left( 7 \right).
Hence, evaluate g \left( f \left( 7 \right) \right).
Find g \left( 7 \right).
Hence, evaluate f \left( g \left( 7 \right) \right).
Does f \left( g \left(x\right)\right) = g \left( f \left(x\right)\right) for all x?
Consider the following functions:
\begin{aligned} f \left( x \right) &= - 2 x - 8 \\ g \left( x \right) &= 4 x^{2} - 4 \end{aligned}Evaluate g \left( f \left( 6 \right) \right).
If h \left( x \right) is defined as f \left( g \left( x \right) \right), state the equation for h \left( x \right).
What type of functions are f \left( g \left( x \right) \right) and g \left( f \left( x \right) \right)?
Consider the following functions:
\begin{aligned} f \left(x\right) &= - 2 x + 2 \\ g \left(x\right) &= 4 x^{2} - 8 \\ r \left(x\right) &= - 3 x - 8 \end{aligned}Find g \left( 6 \right).
Hence, evaluate f \left( g \left( 6 \right) \right).
Hence, evaluate r \left( f \left( g\left( 6 \right) \right)\right).
Consider the following functions:
\begin{aligned} f \left(x\right) &= - 2 x + 6 \\ g \left(x\right) &= 3 x + 1 \end{aligned}If r \left(x\right) is defined as f \left(x^{2}\right), state the equation for r \left(x\right).
Hence, state the equation for q \left(x\right), which is g \left( f \left(x^{2}\right)\right).
The function f \left(x\right) is defined as f \left(x\right) = - 3 x + 4 g \left(x\right).
Given that f \left(x\right) is a quadratic function, what type of function is g \left(x\right)?
If f \left(x\right) = -3x + 20x^{2}, find the equation for g\left(x\right).
Find an algebraic expression for the function g \left( f \left(x\right)\right).
Consider the following functions:
\begin{aligned} p \left(x\right) &= x + 3 \\ q \left(x\right) &= x^{2} - 1 \\ r \left(x\right) &= \left(x + 4\right) \left(x + 2\right) \end{aligned}Rewrite r \left(x\right) in expanded form.
Show that r \left(x\right) = q \left( p \left(x\right)\right).
Write an algebraic expression for p \left( q \left(x\right)\right).
Find the range of values for which r \left(x\right) < p \left( q \left(x\right)\right).
Consider the following functions:
\begin{aligned} f \left(x\right) &= x^{2} + 3 \\ g \left(x\right) &= 4 x - 9 \end{aligned}State the equation for f \left( 2 x\right).
Show that f \left( 2 x\right) = g \left( f \left(x\right)\right).
Consider the function f \left(x\right) = 5 x^{2} + 4. Define g \left(x\right) such that g \left( f \left(x\right)\right) = f \left(x\right) for all x.
Consider the following functions:
\begin{aligned} h \left( x \right) &= \left(x - 6\right)^{3} - 3 \left(x - 6\right)^{2} + 8 \left(x - 6\right) - 5 \\ g \left( x \right) &= x - 6 \end{aligned}Find f \left( x \right) such that h(x) = \left(f\ \circ\ g\right)(x).
For each of the following pairs of functions, sketch (f \circ g)(x) on a number plane:
f(x)=x and g(x)=\sin x
f(x)=x^3 and g(x)=x-5
f(x)=x+1 and g(x)=-x^2
f(x)=e^x and g(x)=\ln x
f(x)=\ln x and g(x)=e^x
f(x)=e^x and g(x)=\ln (x-1)^2
f(x)=\cos x and g(x)=2x
Consider a square with a side length of x units.
Write an equation for the perimeter, p, of the square in terms of x.
Write the area, A, in terms of the perimeter p.
Calculate the area of a square that has a perimeter of 22 units.
A conical container is being filled with water, as shown in the diagram. The water is being poured in such a way that the radius of the water's surface increases at a rate of 7\text{ cm/s}.
Find a function r \left( t \right) for the radius after t seconds.
Find a function A \left( r \right) for the area of the water's surface in terms of the radius r.
Find the composite function \left(A\ \circ\ r\right)(t).
Interpret the function \left(A\ \circ\ r\right)(t) in context.
Air is being added to a spherical balloon. At a time t (in seconds), the radius r of the balloon (in centimetres) can be given by the function r \left( t \right) = 3 \sqrt{t}. The volume of a sphere in terms of its radius is given by the formula V \left( r \right) = \dfrac{4}{3} \pi r^{3}.
Find the composite function (V \circ r ) \left( t \right).
Interpret the function \left(V \circ r \right) \left(t\right) in context.
Calculate the volume of the balloon after 6 seconds. Give your answer correct to two decimal places.
During a sale at a certain clothing store, all shirts are on sale for \$10 less than 75\% of the original price, x.
Write a function g that finds 75\% of x.
Write a function f that finds 10 less than x.
Construct the composite function \left( f \circ g \right) \left( x \right) =f \left( g \left( x \right) \right).
Hence, calculate the sale price of a shirt that has an original price of \$86.
If the discounts were applied in the other order, using \left(g\circ f\right)\left(x\right), would the sale price of the same shirt increase or decrease?
Moon A has a radius of R and its volume is calculated by the function V \left( R \right) = \dfrac{4}{3} \pi R^{3}.
Construct a model D \left( R \right) that describes the difference between the volumes of Moon A and Moon B, where Moon B has a radius that is 4 times larger than that of Moon A.
How much larger is the volume of Moon B than the volume of Moon A if Moon A has a radius of 13 \text{ km}?
A computer manufacturer sells hard drives to a retail outlet, each at a cost of \$11 more than the manufacturing cost. The retail store then sells each hard drive to the public, charging 60\% more than they paid to the manufacturer.
If m represents the manufacturing cost (in dollars), find a function A \left( m \right) which returns the price of buying a hard drive at the retail store.
Find the price of buying a hard drive at the store, if the manufacturing cost is \$22.50 .
Let m \left( s \right) represent the number of bricks required to construct a building with a surface area s, and let c \left( b \right) represent the cost of b bricks.
Describe what \left(c\ \circ\ m\right)(s) represents in context.
A cylindrical tank initially contains 100 \text{ cm}^3 of oil and starts being filled at a constant rate of 30 \text{ cm}^3 \text{/second}. The radius of the tank is 6 \text{ cm}. Let g be the amount of oil in the container after t seconds.
State the equation for h \left( g \right), the height of the oil in the container, in terms of g.
State the equation for g \left( t \right), the amount of oil in the tank after t seconds.
The function A \left( t \right) is defined as A \left( t \right) = h \left( g \left( t \right) \right). Form an equation for A \left( t \right) in terms of t.
What does A \left( t \right) represent?
Sketch the graph of A \left( t \right).
Describe how quickly the height of the oil in the tank is increasing.
The number of customers at Michael's Corner Store over a period of five days is shown on the following graph:
F \left( x \right) represents the number of female customers.
M \left( x \right) represents the number of male customers.
T \left( x \right) represents the total number of customers.
Evaluate \left( T - M \right) \left( 2 \right).
Describe what \left( T - M \right) \left( 2 \right) represents in context.
Evaluate \left( T - F \right) \left( 4 \right).
Describe what \left( T - F \right) \left( 4 \right) represents in context.
Determine whether the following statements are true or false:
The number of female customers increased from Day 2 to Day 3.
The number of male customers decreased from Day 4 to Day 5.
The total number of customers did not change from Day 2 to Day 3.
There were more female customers than male customers on most days.
The total number of customers decreased from Day 4 to Day 5.
The price of a stock at the end of the nth trading day is given by S(n) = \dfrac{1}{n+1}. A trader decides to purchase stocks, and the number of stocks they own at the end of the nth trading day is approximated by B(n) = (n + 1)^2.
Find the equation for the value, V(n), of the trader’s stocks at day n.
Sketch the graph of V(n).
An employer’s revenue at time t is given by R(t) = 1000e^t. The employer must pay their staff wages, which is typically 70\% of the employer’s revenue.
Find the equation for the total wages the employer must pay, W(t).
Find the equation for the employer's profit, P(t).
Sketch the graph of the employer’s profit over time on a number plane.
An oil spill has a radius that grows according to r(t) = t + 5, where t is the number of minutes passed since the spill.
Find the equation of the area of the oil spill, A(t).
Sketch the graph of A(t) on a number plane.
The number of bacteria increases according to N(t) = 2^t, where t is the number of minutes pass. The area of a single bacteria is about 3 square micrometres.
Find the area, A, of the bacteria after t minutes.
Sketch the graph of the area, A, after t minutes.
The birth rate of a population is b(t) = 3t^2 and the death rate is d(t) = t.
Find the equation for the net rate, n(t)=b(t) - d(t), of the population at time t.
Sketch the graph of the net rate of the population as a function of time t.