4.03 Function notation
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 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).