Demonstration 1 of 4
Exactly one output
Which of these relationships keep the promise of one output for each input?
Count the arrows leaving each input. Two inputs sharing one output is fine. One input with two outputs breaks the condition, as in Panel C and in y² = x.
\[y^2 = x\]
\[y = x^2\]
Inputs sit on the left and outputs on the right. Each arrow is one assignment. A function gives every allowed input exactly one arrow.
Predict first. Panel B sends two different inputs to 4. Is it still a function?
Choose an example
Constructed example: the chapter's Figure 16.1 panels and its equation y² = x.
Calculated values
- Arrows
- -2 to 4; 0 to 0; 2 to 4
- Inputs with two outputs
- none
- Outputs shared by two inputs
- 4
- Function?
- yes
Count the arrows leaving each input: 1 + 1 + 1 = 3 arrows for 3 inputs. (-2)² = 4 and 2² = 4, so two inputs share output 4. That is allowed: each input still has exactly one arrow, so this is a function.
Use the idea
When a table or rule gives you outputs, check each input separately: one destination each, and no allowed input left out.
Where the conclusion applies
The domains are the inputs shown. For y² = x, the inputs are 0, 1 and 4. A relationship that fails the test is still a valid relationship, just not a function in this direction.
Check your understanding: Is y = x² a function on the inputs -1, 0, 1?
Chapter 16 source: section "The one-output condition". Demonstration C16-D01.
Demonstration 2 of 4
Root first, then power
What does a fractional exponent such as 2/3 ask you to do?
The curve shows the base raised to every exponent from 0 to 1.25. The dot is the true power; the cross is what you get by mistakenly multiplying the base by the exponent.
\[8^{2/3}=(8^{1/3})^2=4\]
\[4^{1/2}4^{1/2}=4^1\]
In a fractional exponent, the denominator names the root (2 for square root, 3 for cube root) and the numerator counts the power. The base is positive.
Predict first. Will 8 to the power 1/2 be a whole number, or only approximately a decimal?
Choose an example
Constructed example: the chapter's fractional powers of 4 and 8.
Calculated values
- Base
- 8
- Exponent
- 2/3
- Cube root of 8
- 2
- Value
- 4
- 2/3 x 8 (the trap)
- 16/3
The denominator 3 asks for the cube root; the numerator 2 counts the power. 8^(2/3) = (8^(1/3))^2 = 2^2 = 4. Check: 4^3 = 64 and 8^2 = 64. Taking 2/3 of 8 would give 16/3, a different number: an exponent does not name that multiplication.
Use the idea
For a fractional power, find the root before applying the numerator's power, then check by raising the answer to the denominator.
Where the conclusion applies
Positive bases only: a negative number has no real square root. Values that are not exact are rounded to four decimals and called approximate.
Check your understanding: What is 4 to the power 3/2?
Chapter 16 source: section "Fractional inputs need fractional powers". Demonstration C16-D02.
Demonstration 3 of 4
Second differences
How can a table at equally spaced inputs reveal a quadratic pattern?
The dots are the outputs at 0 to 4. The blue row lists first differences; they grow. The bold row lists second differences; they stay at 2a.
\[q(x)=2x^2+1\]
\[ax^2+bx+c\]
First differences subtract neighboring outputs. Second differences subtract neighboring first differences. For ax² + bx + c at unit steps, the second difference is 2a.
Predict first. If the 2 in 2x² + 1 becomes 3, what will every second difference be?
Choose an example
Constructed example: the chapter's q(x) = 2x² + 1, with other leading coefficients.
Calculated values
- Rule
- q(x) = 2x² + 1
- Outputs at 0 to 4
- 1, 3, 9, 19, 33
- First differences
- 2, 6, 10, 14
- Second differences
- 4, 4, 4
- 2a
- 2 x 2 = 4
- Quadratic?
- yes
Outputs 1, 3, 9, 19, 33. First differences: 3 - 1 = 2, 9 - 3 = 6, and so on: 2, 6, 10, 14. Second differences: 4, 4, 4, each equal to 2a = 2 x 2 = 4. Constant, nonzero second differences at equal input steps fit a quadratic.
Use the idea
To test a table for a quadratic pattern, check that the inputs are equally spaced, then take differences twice.
Where the conclusion applies
A finite table supports a pattern choice but cannot prove an unknown process follows the rule at unlisted inputs. With a = 0 the rule is not quadratic.
Check your understanding: For q(x) = x² + 1 at inputs 0 to 4, what are the outputs and the second differences?
Chapter 16 source: section "Worked example: second differences identify a quadratic pattern". Demonstration C16-D03.
Demonstration 4 of 4
Three families
What separates a linear, a quadratic and an exponential pattern?
The highlighted curve is the chosen family; the others stay in grey for comparison. The row under the graph shows each step's difference or factor.
\[f(x) = 2x + 3\]
\[g(x) = x^2\]
\[h(x) = 2^x\]
A difference subtracts one output from the next. A factor divides the next output by the one before. The inputs step by 1 from -2 to 2.
Predict first. For h(x) = 2^x, which stays constant from step to step: the difference or the factor?
Choose an example
Constructed example: the chapter's table for f, g and h at inputs -2 to 2.
Calculated values
- Rule
- f(x) = 2x + 3
- Outputs at -2 to 2
- -1, 1, 3, 5, 7
- Differences
- 1 - (-1) = 2; 3 - 1 = 2; 5 - 3 = 2; 7 - 5 = 2
- Constant difference?
- yes
For f(x) = 2x + 3, the outputs at -2, -1, 0, 1, 2 are -1, 1, 3, 5, 7. Each difference: 1 - (-1) = 2; 3 - 1 = 2; 5 - 3 = 2; 7 - 5 = 2. The difference is the same every step, the mark of a linear pattern.
Use the idea
Given a column of outputs at equal steps, try both: a repeated difference points to linear, a repeated factor points to exponential.
Where the conclusion applies
Each function is defined for all real numbers; the table samples five inputs. A factor is undefined when the earlier output is 0, as for g(x) = x² at x = 0.
Check your understanding: For f(x) = 2x + 3, what is the factor from x = 1 to x = 2, and is it the same as from 0 to 1?
Chapter 16 source: section "Three families, three kinds of change". Demonstration C16-D04.