For the following functions, determine the output produced by the indicated input value:
f \left(x\right) = - 6 x + 4; x = 4
f \left(x\right) = 9 x^{2} + 7 x - 4; x = - 4
g \left(t\right) = - \dfrac{2}{t} + 8; t = - 8
h \left(t\right) = 9^{t} + 6; t = 2
Rewrite the following statements using function notation:
Suppose that \left(7, - 6 \right) is an ordered pair that satisfies the function g.
In the equation x - 4 y = 8, y is a function of x.
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 graph of y = f \left( x \right):
Determine the function value at x = 3.
Is the graph continuous at x = 3?
Consider the function h \left(x\right) = - x^{2} + 6 x - 6.
For each input value of x, state the maximum number of distinct output values h(x) can produce.
Is h \left(x\right) a function?
For each function, find the function value as indicated. Round your answers to two decimal places where necessary.
Consider the function f(x) = x^{2} - 49. If f(x) = 12, find the values of x. Round your answer to two decimal places.
Consider the function f \left( x \right) = \left(x + 4\right) \left(x^{2} - 4\right). If f \left( x \right) = 0, find the values of x.
Consider the function Z(y) = y^{2} + 12 y + 32. If Z(y) = - 3, find the values of y.
Form an expression for each of the following:
If A \left( x \right) = x^{2} + 1 and Q \left( x \right) = x^{2} + 9 x, evaluate:
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. Does f \left( a \right) + f \left( b \right) = f \left( a + b \right) for all values of a and b? Show working to justify your answer.
Suppose f and g are functions. Find the corresponding point on the graph of y = g \left( x \right), if:
\left(9, 12\right) is a point on the graph of y = f \left( x \right) and g \left( x \right) = f \left( x \right) - 8.
\left(6, - 7 \right) is a point on the graph of y = f \left( x \right) and g \left( x \right) = f \left( x - 5 \right).
\left(9, - 12 \right) is a point on the graph of y = f \left( x \right) and g \left( x \right) = 6 f \left( x \right).
Given the table of values, find \left(f + g\right)\left(2\right).
| x | 2 | 7 | 8 | 9 |
|---|---|---|---|---|
| f(x) | 4 | 14 | 18 | 16 |
| g(x) | 8 | 28 | 36 | 32 |
Given the table of values, find (f\times g)\left(6\right).
| x | 2 | 5 | 6 | 9 |
|---|---|---|---|---|
| f(x) | 4 | 10 | 12 | 18 |
| g(x) | 8 | 20 | 24 | 36 |
Let f \left( x \right) = x^{2} + 6 and g \left( x \right) = 5 x - 3.
Find \left(f - g\right) \left(x\right).
Evaluate \left(f - g\right)\left(5\right)
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)
For each of the following functions, find:
For the function f \left( x \right) = x^{2} + 2 x, find \dfrac{f \left( 6 + h \right) - f \left( 6 \right)}{h}.
For the function f \left( x \right) = x^{2} + 3 x, find \dfrac{f \left( x + h \right) - f \left( x \right)}{h}.
For the function f \left( x \right) = x^{2} + 2 x, find \dfrac{f \left( p \right) - f \left( q \right)}{p - q}.
The financial team at Kerzon Corp. wants to calculate the profit (in dollars), P \left( x \right), generated by producing x units of personalised stationery.
The revenue (in dollars) produced by the product is given by the equation is R \left( x \right) = - \dfrac{x^{2}}{4} + 50 x. The cost of production (in dollars) is given by the equation C \left( x \right) = 12 x + 14.
The profit is calculated as P \left( x \right) = R \left( x \right) - C \left( x \right).
Find an equation for P \left( x \right) in terms of x.
Find the values of the following:
The financial team at The Gamgee Cooperative wants to calculate the profit (in dollars), P \left( x \right), generated by producing x units of wetsuits.
The revenue (in dollars) produced by the product is given by the equation is R \left( x \right) = - \dfrac{x^{2}}{4} + 40 x. The cost of production (in dollars) 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 equation for P \left( x \right) in terms of x.
Find the values of the following:
Sketch the graphs of y = R \left( x \right), y = C \left( x \right) and y = P \left( x \right) on the same number plane.