The point \left(7, - 6 \right) satisfies the function f(x). Write this using function notation.
If f \left(x\right) = 9 x^{2} + 7 x - 4, find:
f(-4)
f(10)
If f \left(x\right) = - 6 x + 4, find:
f \left( 4 \right)
f \left( 0 \right)
If f(x) = 4 x + 4, find:
f \left( 2 \right)
f \left( - 5 \right)
If f \left( x \right) = 3 x - 1, find:
f \left( 3 \right)
f \left( - 4 \right)
If g \left( x \right) = \dfrac{7 x}{4}, find:
g \left( 5 \right)
g \left( - 4 \right)
Consider the function f \left( x \right) = 4 + x^{3}.
Evaluate f \left( 4 \right)
Evaluate f \left( - 2 \right)
Consider the function f \left( x \right) = 2 x^{2} - 2 x + 5. Evaluate f\left(\dfrac{1}{2}\right).
Consider the function f \left( x \right) = \sqrt{ 5 x + 9}. Find the exact value of:
f \left( 0 \right)
f \left( 2 \right)
f \left( - 1 \right)
If f \left( t \right) = \dfrac{t^{3} + 27}{t^{2} + 9} , evaluate the following:
f \left( - 3 \right)
f \left( 3 \right)
f \left( 4 \right)
Consider the function f \left( x \right) = x^{2} + 8 x. Write an expression for:
f \left( a \right)
f \left( b \right)
Consider the function f \left( x \right) = 2 x^{3} + 3 x^{2} - 4.
Evaluate f \left( 0 \right).
Evaluate f \left( \dfrac{1}{4} \right).
If j(x) = 3^{x} - 3^{ - x }, find the following, rounding your answers to two decimal places if necessary:
j(0)
j(1)
j(4)
If m \left( x \right) = \sqrt{12^{2} - x^{2}}, find:
m \left( 4 \right)
m \left( 0 \right)
m \left( 9 \right)
m \left( \sqrt{2} \right)
Consider the function p \left( x \right) = x^{2} + 8.
Evaluate p \left( 2 \right).
Form an expression for p \left( m \right).
Consider the equation x + 3 y = 6.
Rewrite the equation using function notation f \left( x \right) for y.
Find the value of f \left( 3 \right).
Consider the equation x - 4 y = 8.
Rewrite the equation using function notation f \left( x \right).
Find the value of f \left( 12 \right).
Consider the equation y + 6 x^{2} = 3 - x.
Rewrite the equation using function notation f \left( x \right).
Find the value of f \left( 3 \right).
Consider the equation y - 4 x^{2} = 5 + x.
Rewrite the equation using function notation f \left( x \right).
Find the value of f \left( 2 \right).
Consider the equation - 6 x + 5 y = 7.
Rewrite the equation using function notation f \left( x \right).
Find the value of f \left( 3 \right).
Use the graph of the function f \left( x \right) to find each of the following values:
f \left( 0 \right)
f \left( - 2 \right)
The value of x such that f \left( x \right) = 3
Consider the relation h \left(x\right) = - x^{2} + 6 x - 6.
For any input value of x, state the maximum number of distinct output values h \left(x\right) can produce.
Is h \left(x\right) a function? Explain your answer.
For each of the following pairs of variables, determine which is the independent variable and which is the dependent variable:
Cost of pizza and size of pizza.
Mark achieved on a test and time spent studying.
Duration of a loan and amound borrowed.
Time spent exercising and physical fitness level.
Consider the function f(x) = x^{2} - 49.
Find:
f(1)
f(8)
f(0)
If f(x) = 12, what are the possible values for x, rounded to two decimal places?
Consider the function g \left( x \right) = a x^{3} - 3 x + 5.
Form an expression for g \left( k \right).
Form an expression for g \left( - k \right).
Is g \left( k \right) = g \left( - k \right)?
Is g \left( k \right) = - g \left( - k \right)?
A function is defined as f \left( x \right) = 2 + 3 x - x^{3}. Evaluate f \left( - 2 \right) + f \left( 4 \right).
If A \left( x \right) = x^{2} + 1 and Q \left( x \right) = x^{2} + 9 x, evaluate the following:
A \left( 5 \right)
Q \left( 4 \right)
A \left( 3 \right) + Q \left( 2 \right)
Consider the function f \left( x \right) = x^{3} - 8 x + 7.
Evaluate f \left( 4 \right).
Evaluate f \left( - 4 \right).
Form an expression for f \left( a \right).
Form an expression for f \left( b \right).
Form an expression for f \left( a + b \right).
Is f \left( a \right) + f \left( b \right) = f \left( a + b \right) for all values of a and b?
Consider the function f \left( x \right) = x^{2} + 5 x. Find f(a + h).
Consider the function f \left( x \right) = x^{2} - 2 x. Find f(a + 6).
Consider the function f \left( x \right) = x^{2} - 3 x - 2.
Form an expression for f \left( a - 2 \right).
Form an expression for f \left( a + h \right) - f \left( a \right).
Consider the function f \left( x \right) = 4 x^{2} + 4 x + 3.
Form an expression for f \left( a + 2 \right).
Form an expression for f \left( a + h \right) - f \left( a \right).
Consider the following table of values:
Find (f + g)(2).
Find (f + g)(7).
Find (f \times g)(2).
Find (g-f)(9).
| x | 2 | 7 | 8 | 9 |
|---|---|---|---|---|
| f(x) | 4 | 14 | 16 | 18 |
| g(x) | 8 | 28 | 32 | 36 |
Let f \left( x \right) = x^{2}-1 and g \left( x \right) = 5 x-1. Find f(2) + g(2).
Let f \left( x \right) = - 5 x + 3 and g \left( x \right) = x^{2} - 7.
Find \left(f \times g\right)\left(x\right).
Evaluate \left(f\times{g}\right)\left( - 3 \right).
If f(x) = 3 x - 5 and g(x) = 5 x + 7, find:
(f+g)(x)
(f+g)\left(4\right)
(f-g)(x)
(f-g)\left(10\right)
If f \left( x \right) = 4 x^{3} and g \left( x \right) = x + 5, find:
Consider the functions f \left( x \right) = - 2 x - 6 and g \left( x \right) = 5 x - 7.
Find f \left( 7 \right).
Hence, or otherwise, evaluate g \left( f \left( 7 \right) \right).
Find g \left( 7 \right).
Hence, evaluate f \left( g \left( 7 \right) \right).
Is it true that f \left( g \left(x\right)\right) = g \left( f \left(x\right)\right) for all x?
Consider the functions f \left( x \right) = - 2 x - 8 and g \left( x \right) = 4 x^{2} - 4.
Evaluate (g \circ f )\left( 6 \right) .
The function h \left( x \right)= (f \circ g ) \left( x \right). Write h \left( x \right) in terms of x.
Are (f \circ g )\left(x\right) and (g \circ f ) \left(x\right) both quadratic functions?
Consider the functions f \left(x\right) = - 2 x + 2, g \left(x\right) = 4 x^{2} - 8 and r \left(x\right) = - 3 x - 8.
Evaluate (f \circ g) \left(6\right).
Evaluate (r \circ f \circ g) \left(6\right).
Consider f \left(x\right) = x^{2} + 3 and g \left(x\right) = 4 x - 9.
Define f \left( 2 x\right).
Show that f \left( 2 x\right) = g \left( f \left(x\right)\right).
If Z(y) = y^{2} + 12 y + 32, find y when Z(y) = - 3.
A function f \left( x \right) is defined by f \left( x \right) = \left(x + 4\right) \left(x^{2} - 4\right).
Evaluate f \left( 6 \right).
Find all solutions for which f \left( x \right) = 0.
A graph of a quadratic equation of the form y = a x^{2} + b x + c passes through the points \left(0, - 7 \right), \left( - 1 , - 8 \right) and \left(4, 77\right) .
Using the point \left(0, - 7 \right), find the value of c.
Substitute c and \left( - 1 , - 8 \right) into the equation y = a x^{2} + b x + c to obtain an equation that describes a in terms of b.
Substitute c and \left(4, 77\right) into the equation y = a x^{2} + b x + c to obtain a simplified second equation in terms of a and b.
Solve for a and b.
The financial team at The Gamgee Cooperative wants to calculate the profit, P \left( x \right), generated by producing x units of wetsuits.
The revenue produced by the product is given by the equation is R \left( x \right) = - \dfrac{x^{2}}{4} + 40 x. The cost of production is given by the equation C \left( x \right) = 5 x + 410.
The profit is calculated as P \left( x \right) = R \left( x \right) - C \left( x \right).
Find an expression for P \left( x \right) in terms of x.
Find the values of the following:
R \left( 70 \right)
C \left( 70 \right)
P \left( 70 \right)
Sketch the graphs of y = R \left( x \right), y = C \left( x \right) and y = P \left( x \right).
Elizabeth wants to calculate the cost to travel to Tehran and then Mexico City at certain times of the year. A program on her computer calculates T \left( x \right), the total cost of this trip, by adding the cost of travelling to Tehran, S \left( x \right), and the cost of travelling to Mexico City, U \left( x \right).
Based on historical data, the computer program uses the calculations:
S \left( x \right) = x^{2} - 200 x + 10\,227 \quad \text{ and } \quad U \left( x \right) = 18 x + 15\,423 where x is the number of days until the date of travel.
Find an equation for T \left( x \right).
Find the values of the following:
S \left( 91 \right)
U \left( 91 \right)
T \left( 91 \right)
Sketch the graphs of y = S \left( x \right), y = U \left( x \right) and y = T \left( x \right).
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 \text{cm}) 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?