3.1 Functions and Function Notation

In a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? the table shows a possible rule for assigning grade points.

the tablehttp://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014. lists the five greatest baseball players of all time in order of rank.

1. ⓐIs the rank a function of the player name?
2. ⓑIs the player name a function of the rank?

A function $N = f \left(y\right)$ gives the number of police officers, $N ,$ in a town in year $y .$ What does $f \left(2005\right) = 300$ represent?

Use function notation to express the weight of a pig in pounds as a function of its age in days $d .$

Which table, the table, the table, or the table, represents a function (if any)?

Does the table represent a function?

Evaluate $f \left(x\right) = x^{2} + 3 x - 4$ at:

1. ⓐ $2$
2. ⓑ $a$
3. ⓒ $a + h$
4. ⓓ Now evaluate $\frac{f \left(a + h\right) - f \left(a\right)}{h}$

Given the function $h \left(p\right) = p^{2} + 2 p ,$ evaluate $h \left(4\right) .$

Given the function $g \left(m\right) = \sqrt{m - 4} ,$ evaluate $g \left(5\right) .$

Given the function $g \left(m\right) = \sqrt{m - 4} ,$ solve $g \left(m\right) = 2.$

Express the relationship $2 n + 6 p = 12$ as a function $p = f \left(n\right) ,$ if possible.

Does the equation $x^{2} + y^{2} = 1$ represent a function with $x$ as input and $y$ as output? If so, express the relationship as a function $y = f \left(x\right) .$

If $x - 8 y^{3} = 0 ,$ express $y$ as a function of $x .$

Using the table,
ⓐ Evaluate $g \left(3\right) .$
ⓑ Solve $g \left(n\right) = 6.$

Using the table from Evaluating and Solving a Tabular Function above, evaluate $g \left(1\right) .$

Using the figure, solve $f \left(x\right) = 1.$

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

1. ⓐ Is a balance a function of the bank account number?
2. ⓑ Is a bank account number a function of the balance?
3. ⓒ Is a balance a one-to-one function of the bank account number?

Evaluate the following:
ⓐ If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?
ⓑ If so, is the function one-to-one?

Is the graph shown in the figure(c) one-to-one?

What is the difference between a relation and a function?

Why does the vertical line test tell us whether the graph of a relation represents a function?

Why does the horizontal line test tell us whether the graph of a function is one-to-one?

Determine whether the relation represents a function.

$\left\{\left(a , b\right) , \left(b , c\right) , \left(c , c\right)\right\}$

Determine whether the relation represents $y$ as a function of $x .$

$y = x^{2}$

Determine whether the relation represents $y$ as a function of $x .$

$3 x^{2} + y = 14$

Determine whether the relation represents $y$ as a function of $x .$

$y = - 2 x^{2} + 40 x$

Determine whether the relation represents $y$ as a function of $x .$

$x = \frac{3 y + 5}{7 y - 1}$

Determine whether the relation represents $y$ as a function of $x .$

$y = \frac{3 x + 5}{7 x - 1}$

Determine whether the relation represents $y$ as a function of $x .$

$2 x y = 1$

Determine whether the relation represents $y$ as a function of $x .$

$y = x^{3}$

Determine whether the relation represents $y$ as a function of $x .$

$x = \pm \sqrt{1 - y}$

Determine whether the relation represents $y$ as a function of $x .$

$y^{2} = x^{2}$

Evaluate $f \left(−3\right) , f \left(2\right) , f \left(− a\right) , − f \left(a\right) , f \left(a + h\right) .$

$f \left(x\right) = 2 x - 5$

Evaluate $f \left(−3\right) , f \left(2\right) , f \left(− a\right) , − f \left(a\right) , f \left(a + h\right) .$

$f \left(x\right) = \sqrt{2 - x} + 5$

Evaluate $f \left(−3\right) , f \left(2\right) , f \left(− a\right) , − f \left(a\right) , f \left(a + h\right) .$

$f \left(x\right) = \left|x - 1\right| - \left|x + 1\right|$

Evaluate $f \left(−3\right) , f \left(2\right) , f \left(− a\right) , − f \left(a\right) , f \left(a + h\right) .$

Given the function $g \left(x\right) = x^{2} + 2 x ,$ evaluate $\frac{g \left(x\right) - g \left(a\right)}{x - a} , x \neq a .$

Evaluate $f \left(−3\right) , f \left(2\right) , f \left(− a\right) , − f \left(a\right) , f \left(a + h\right) .$

Given the function $f \left(x\right) = 8 - 3 x :$

1. ⓐ Evaluate $f \left(- 2\right) .$
2. ⓑ Solve $f \left(x\right) = - 1.$

Evaluate $f \left(−3\right) , f \left(2\right) , f \left(− a\right) , − f \left(a\right) , f \left(a + h\right) .$

Given the function $f \left(x\right) = x^{2} - 3 x :$

1. ⓐ Evaluate $f \left(5\right) .$
2. ⓑ Solve $f \left(x\right) = 4.$

Evaluate $f \left(−3\right) , f \left(2\right) , f \left(− a\right) , − f \left(a\right) , f \left(a + h\right) .$

Consider the relationship $3 r + 2 t = 18.$

1. ⓐ Write the relationship as a function $r = f \left(t\right) .$
2. ⓑ Evaluate $f \left(- 3\right) .$
3. ⓒ Solve $f \left(t\right) = 2.$

Determine whether the relation represents a function.

$\left\{\left(3 , 4\right) , \left(4 , 5\right) , \left(5 , 6\right)\right\}$

Determine if the relation represented in table form represents $y$ as a function of $x .$

Determine if the relation represented in table form represents $y$ as a function of $x .$

Solve $f \left(x\right) = 1.$

Evaluate the function $f$ at the values $f \left(- 2\right) , f \left(- 1\right) , f \left(0\right) , f \left(1\right) ,$ and $f \left(2\right) .$

$f \left(x\right) = 8 - 3 x$

Evaluate the function $f$ at the values $f \left(- 2\right) , f \left(- 1\right) , f \left(0\right) , f \left(1\right) ,$ and $f \left(2\right) .$

$f \left(x\right) = 3 + \sqrt{x + 3}$

Evaluate the function $f$ at the values $f \left(- 2\right) , f \left(- 1\right) , f \left(0\right) , f \left(1\right) ,$ and $f \left(2\right) .$

$f \left(x\right) = 3^{x}$

$f \left(x\right) = 3 x - 2$
$g \left(x\right) = 5 - x^{2}$
$h \left(x\right) = - 2 x^{2} + 3 x - 1$

$f \left(\frac{7}{3}\right) - h \left(- 2\right)$

The number of cubic yards of dirt, $D ,$ needed to cover a garden with area $a$ square feet is given by $D = g \left(a\right) .$

1. ⓐ A garden with area 5000 ft² requires 50 yd³ of dirt. Express this information in terms of the function $g .$
2. ⓑ Explain the meaning of the statement $g \left(100\right) = 1.$

Let $h \left(t\right)$ be the height above ground, in feet, of a rocket $t$ seconds after launching. Explain the meaning of each statement:

1. ⓐ $h \left(1\right) = 200$
2. ⓑ $h \left(2\right) = 350$