4.1E Exercises

Describing Functions

Describe in words what instructions are being given in the algebraic definitions of functions below.

1. \( g(x) = x^2 - 1 \)
2. \( K(m) = \frac{8}{5}m \)
3. \( A(r) = \pi r^2 \)
4. \( h(x) = \dfrac{3x+1}{5} \)
5. \( f(x) = |x| = \begin{cases} x & \text{ if } x\geq 0 \\ -x &\text{ if } x< 0 \end{cases} \)
6. \( s(t) = 6 + 3t - 4.9t^2 \)
7. \( P(u) = 1300u - (1.5u^2 + 30u) \)
8. \( F(C) = \frac{9}{5} C + 32 \)

Answer

This is all in your own words, however makes sense to you, AS LONG AS it is correct in the order of operations happening!

1. Square the input, then subtract 1.
2. Multiply the input by \( \frac{8}{5} \). This function approximates a conversion from speed in miles per hour to speed in kilometers per hour.
3. Square the input, then multiply by \( \pi\). This function computes the area of a circle depending on its radius.
4. Multiply the input by 3, then add 1, then divide the result by 5.
5. Take the absolute value of the input; that is, if the input is nonnegative, return the exact input, but if it is negative, return its opposite.
6. Multiply the input by 3 and add it to 6, then subtract from that the result of "squaring the input and then multiplying by 4.9".
7. Square the input, multiply by 1.5, and add the result of "multiplying 30 by the input." Subtract all of that from the input multiplied by 1300. This could be a profit function depending on number of units sold, \(u\), which subtracts a formula for costs from a formula calculating revenue.
8. Multiply the input by \(\frac{9}{5}\) and then add 32. This function takes a Celsius temperature as input and converts it to the Fahrenheit equivalent.

Evaluating Functions

Evaluate the functions at the indicated inputs. Report your answers with appropriate function notation!

1. \( g(x) = x^2 - 1 \) at \(x = -2, 0, 5, \) and \( x+h \).
2. \( K(m) = \frac{8}{5}m \) at \(m = 0, 25, 45,\) and \(75\).
3. \( H(x) = \dfrac{3x+1}{5} \) at \(x = 0, \frac{1}{3}, 3,\) and \( x+h\).
4. \(s(t) = 6 + 3t - 4.9t^2 \) at \( t = 0, 1, \) and \(t_f \).
5. (Calculator allowed) \( P(u) = 1300u - (1.5u^2 + 30u) \) at \( u = 2, 10,\) and \(\blacksquare\).
6. \( F(C) = \frac{9}{5} C + 32 \) at \( C = 0, 20,\) and \(100\).

Answer

1. \( g(-2) = 3,\) \( g(0) = -1 \), \( g(5) = 24 \), \( g(x+h) = (x+h)^2 - 1 \)
2. \( K(0) = 0, K(25) = 40, K(45) = 72, K(75) = 120 \)
3. \( H(0) = \frac{1}{5}, H(1/3) = \frac{2}{5}, H(3) = 2, H(x+h) = \dfrac{3(x+h) +1}{5} \)
4. \( s(0) = 6, s(1) = 4.1, s(t_f) = 6 + 3t_f - 4.9 t_f^2 \)
5. \( P(2) = 2534, P(10) = 12550, P(\blacksquare) = 1300\blacksquare - (1.5\blacksquare^2 + 30\blacksquare) \)
6. \( F(0) = 32, F(20) = 68, F(100) = 212\)

Difference Quotients

A difference quotient is an expression you will see roughly 2130329028 times in Calc I of the form \( \dfrac{ f(x+h) - f(x)}{h} \). For each function in particular, write the difference quotient and simplify if reasonable. Be careful when plugging in, when subtracting, and when cancelling! For example, \(f(x) = x^2+ 1\) would yield

\[ \dfrac{ (x+h)^2+ 1 - (x^2+1)}{h} = \dfrac{2hx+h^2}{h} = 2x+h \notag \]

1. \( f(x) = x^3 \)
2. \( f(x) = 2x + 3 \)
3. \( f(x) = 4 \)

Answer

1. \( \dfrac{ (x+h)^3 - x^3}{h} = \cdots = 3hx+3x^2+h^2 \)

2. \( \dfrac{ 2(x+h)+3 - (2x+3)}{h} = \cdots = 2 \)

3. \( \dfrac{ 4 - 4}{h} = 0 \) (Watch out! There is no variable in the function's defining expression. This is called a constant function, and we'll see it later again.)

Functions From Tables

For each table of values, tell whether it can represent a function, and if so, guess the function that the value pairs would satisfy.

1.

inputs	outputs
-2	2
-1	1
0	0
4	4

2.

inputs	outputs
5	11
6	13
7	15
8	17

3.

inputs	outputs
-1	1
17	4
2	0
-1	9

Answer

1. Yes, it's fine for multiple inputs to be assigned the same output. The function looks like \( f(x) = |x| \).
2. Yes, looks like \( f(x) = 2x + 1 \).
3. No, there is an input sent to two distinct outputs, so this is not a function.

Domain of a Function

Give the domains for the following functions.

1. \( C(x) = x^2 + 40x, \quad 0 \leq x \leq 100 \)
2. \( p(x) = 3x^3 + 2x^2 + x \)
3. (Calculator allowed) \( h(t) = 100 + 3t - 4.9t^2\), which gives the height above the ground, in feet, at time \(t\) seconds, for a rock dropped off a 100ft platform.
4. \( r(x) = \dfrac{3}{x^2 - 5x - 6} \)
5. \( s(x) = \sqrt{x^2 - x} \) (Secretly also an inequality solving problem.)
6. \( f(n) = \dfrac{n+1}{n^2 + 1},\) where \(n = 1, 2, 3, ... \)
7. \( q(x) = x^7 - x^2 + 3 \)
8. \( C(x) = 0.25x^2 - 15x + 2500 \), which gives the total cost of producing \(x\) washing machines.
9. \( g(x) = \dfrac{1}{x^2 - 16} \)
10. \( F(x) = \dfrac{\sqrt{x}}{(x -2)\sqrt{x-1}} \)

Answer

1. The domain is given to us, \( 0 \leq x \leq 100 \). This could be a cost function for producing \(x\) objects where it doesn't make sense to make a negative number of objects and it isn't feasible to make more than 100.
2. All real numbers, \( (-\infty, \infty) \).
3. I start the clock at time 0, as negative time doesn't make sense, outside of Doctor Who I suppose, so \(t \geq 0\). But it also doesn't make sense to continue looking at time values after the rock hits the ground, aka when \(h(t) = 0\)ft. I'm going to stop my domain at \(t = 4.8\) seconds. Figure out why (you can use a calculator and review how to solve a quadratic equation)... Final answer is \( 0 \leq t \leq 4.8\).
4. \( \{x \: | \: x \neq -1,6 \} \)
5. The inside must be nonnegative, and it factors as \( x(x-1)\). There are two ways for this to be 0, when \(x = 0, 1\), and that's fine. There are also two ways this expression can be positive: a) if both \(x > 0\) AND \(x -1 > 0 \), which will be true if \( x> 1\), or b) if both \(x < 0 \) AND \(x - 1 < 0 \), which will happen if \( x < 0\). So my domain includes all real numbers up to and including 0, and then all real numbers greater than or equal to 1. We write this as \( (-\infty, 0] \cup [1, \infty)\).
6. Domain: all natural numbers (positive integers) \(n\).
7. \( (-\infty, \infty) \)
8. \( \{ x \: | \: x \geq 0 \} \) (I can't make negative washing machines... Would that be taking washing machines out back and destroying them with a sledgehammer?)
9. \( \{x \: | \: x \neq \pm 4 \} \)
10. \( \{ x \: | \: x > 1, x \neq 2 \} \)