Find f(x) given x & Find x given f(x)
Given: f(x) = 3x + 1.
Find: f(0), f(1), f(-1) & f(5).
For f(0) → f(0) = 3(0) + 1 → f(0) = 1
For f(1) → f(1) = 3(1) + 1 → f(1) = 4
For f(-1) → f(-1) = 3(-1) + 1 → f(-1) = -2
For f(5) → f(5) = 3(5) + 1 → f(5) = 16
Define a function. Provide an example of one.
A function is a relation where each input has exactly one output.
Example: f(x) = 2x + 3
Identify the slope from the equation:
y = -4x – 3
slope = -4
Find f(0) and the rule:
x: 1 3 5 7
y: 10 18 ? 34
f(0) = 6
f(x) = 4x + 6
f(5) = 4(5) + 6 → 20 + 6 → 26
Identify the y-intercept of y = –3x + 12.
Hint: y-intercept is the y value when when x is = 0. Also know as f(0) or (0, y).
y = -3(0) + 12 → y = 12 → y-intercept = 12.
In other words, (0, 12) is the y-intercept of y = -3x + 12
Given f(x) = –2x + 14 & g(x) = 1/4x - 10
Find f(6) - g(12).
f(6) = -2(6) + 14 → f(6) =2
g(12) = 1/4(12) - 10 → g(12) = -7
f(6) - g(12) = 2 - -7 → f(6) - g(12) = 9
Explain what a constant rate of change means in a linear function.
A constant rate of change means the output changes by the same amount for each equal change in the input.
In a linear function, this ratio is the slope, aka m, aka (y2-y1)/(x2-x1), aka the rise over the run.
Find the slope from the table:
x: 1 3 5 ?
y: 2 6 ? 14
Step 1: find the slope between two points in the table, such as (1, 2) and (3, 6)
Slope m= (y2-y1)/(x2-x1) → (6−2)/(3−1) → 4/2 → 2
Step 2: use the slope to find f(0).
So, when moving -1 from x=1 to x=0, y moves -2 from y=2 to y=0.
Complete the rule: y=2x+0 or y=2x
f(5) = 2(5) → f(5) = 10
(14) = 2x → 7 = x (when y = 14).
Use the table to write a rule and then find f(12)
x: 2 4 6 8
y: 5 ? 17 23
f(0) = -1
f(x) = 3x - 1
f(4) = 3(4) - 1 → 12 - 1 → 11
Using the table below, find the y-intercept of the function.
x: 1 3 5 7
y: 3 9 15 21
y-intercept from table:
x: 1 3 5 7
y: 3 9 15 21
Slope (m) = (15-9)/(5-3) → (6)/(2) → 2
f(0) = 0. (as x increases by 2, y increases by 6. So, x increasing by 1 yields y increasing by 3. Similarly, x decreasing -1 yields y decreasing -3. That means at x = 0, y = 0, because x decreases -1 from 1 to 0 and y decreases -3 from 3 to 0.
y-intercept = 0
Use the table below to find the values of the ?s & the f(6).
x: ? 4 6 8
f(x): 7 11 ? 19
Step 1: find change in f(x)'s rise alongside the change in x's run. Note, the x values go up or down by 2 as y values go up or down by 4. That means x values go up or down by 1 as y values go up or down by 2.
Step 2: find x when f(x) is 7. Note that f(x) moves leftward from 11 to 7, a decrease of 4. From step 1, that means x will decrease by half that amount, from 4 to 2. So, x = 2 when f(x) = 7.
Step 3: find f(x) when x = 6. Note that x moves rightward from 4 to 6, an increase of 2. From step 1, that means f(x) will increase by twice the ammount, from 11 to 15. So, f(6) = 15.
m = 2
f(0) = + 3
Rule = f(x) = 2x + 3
What are the domain and range of a function?
Domain: all possible x-values (inputs)
Range: all possible y-values (outputs)
Find the slope between the points (–2, 5) and (4, 17).
Remember, slope is the ratio between the rate y values change and the rate x values change, also know as the rise over run (Δy/Δx).
Slope between (–2,5) and (4,17)
Slope m = Δy/Δx = (y2-y1)/(x2-x1) →(17-5)/(4--2) → (12)/(6) → 2
Use the table to write the linear rule and find f(15).
x: -2 1 4 10
y: 2 11 20 38
Hints: How much does y change for every one x changes? Also, find f(0).
Answer: f(x) = 3x + 8.
So, f(15) = 3(15) + 8 → f(15) = 45 + 8 → f(15) = 53
Hint follow ups: m = Δy/Δx, or (y2−y1)÷(x2−x1). So, (20-11)÷(4-1) → 9÷3 → 3/1 → 3. So, y rises three units for every one unit x runs.
f(0) = 8
A line has slope 4 and passes through (2, 18).
Find the y-intercept & provide its coordinates.
Given: the line's slope is 4 and it passes through point (2,18).
So, plug (y, x) & m into y=mx+b, then solve for b like so:
18 = 4(2) + b → 18 = 8 + b → 10 = b
y-intercept (b) = 10
y-intercept coordinates = (0,10)
A computer program first runs an incoming data value through the function g(x) = 2x - 1. Then, it routes the output through a second function, f(x) = 4x + 6.
2.) What is the final output for given f(g(5))?
Step 1: Evaluate g(5)
g(5) = 2(5)−1 → 10−1 → 9
Step 2: Substitute g(5) into f(x).
f(g(5)) → f(9) = 4(9)+6 → 36+6 → 42
Answer: f(g(5))=42
What is a relation?
Explain how you can tell if a relation shown on a graph is nonlinear.
A relation is any set of input-output pairs.
A relation is nonlinear if its graph is not a straight line (curves or bends).
A line passes through the points (3, 12) and (9, ?).
The slope is 2.
Find the missing y value.
Step 1: plug in and solve using the slope formula to find the missing y:
m = (y2 - y1)/(x2 - x1)
2 = (y - 12)/(9 - 3) → 2 = (y-12)/6
Step 2: Solve for y.
2 = (y - 12)/6
2 * 6 = y - 12
12 = y - 12
y = 24
A line passes through the points given in the table.
Find the rule, then evaluate f(50).
x: 6 8 12 20
y: 10 9 7 3
Hint: observe how y changes alongside how x does to determine f(0). Then, use f(0) & reverse engineering to determine the rule.
Rule: f(x) = -1/2x + 13
f(50) = -1/2(50)+13 → -25+13 → -12
A line’s graph crosses the y-axis at (0, –7) and also passes through (5, 3).
Write its equation in slope-intercept form (y = mx + b).
Step 1: Find slope: m = (3-(-7)) / (5-0) = (10)/(5) = 2
Step 2: Using y = mx + b, plug in (0, -7) and solve for b. -7 = 2(0) → b = -7
Step 3: Use m and b to write the final equation: y = 2x - 7
Given f(x) = (5/3)x – 12.
Solve for x when f(x) = 18.
Step 1: plug 18 in for f(x) in the equation:18 = (5/3)x - 12
Step 2: Add 12 to both sides: 30 = (5/3)x
Step 3: Multiply by 3/5: x = 30 * 3/5 = 18.
So, x = 18 when f(x) = 18.
What reliable steps can be followed to find the rule of a linear function?
Hint: the rule is an equation that can be used to change any x input to its paired f(x) output.
Hint #2: the equation is of the form f(x) = mx + b, and can be graphed as a straight line.
1.) Find the slope (m) between two given points by calculating the Δy/Δx.
2.) Find the y-intercept (b) by plugging the slope's value and a point's (x, y) values into the the slope-intercept equation for a line, y=mx+b. Then, solve for b.
3.) Rewrite the the slope-intercept equation with values for m & b plugged in and with y & x kept as variables, y = #x + #
You bike up a hill. After 10 min you are at 120 ft elevation. After 25 min, you are at 210 ft.
Find the slope (Δy/Δx) and explain what it means in this context.
Use the value for slope (m), the values from one of your checkpoints (x, y) , and the equation y = mx + b to determine your starting elevation (b).
Step 1: Find and interpret the slope:
m = (210 - 120) / (25 - 10) = 90 / 15 = 6.
This means that the every one minute biked yields +6 feet elevation.
Step 2: Find b using y = mx + b, (x, y) =(10, 120), and m = 6.
120 = 6*10 + b → b = 60
Step 3: plug the values for m & b into the slope-intercept equation: y = 6x + 60.
Write the rule from the linear table and compute f(100).
Find the rule, then evaluate f(50).
x: 6 ? 16 20 22
y: 6 12 ? 27 30
Hint: observe how y changes alongside how x does in the final pairs of points in the table. Apply the same rate of change to find the ?s, f(0), and then the rule.
Rule: f(x) = 3/2x - 3
f(16) = 21
f(100) = 3/2(100) - 3 ➔ 150 - 3 = 147
Given: A function increases by 6 every time x increases by 2.
When x = 10, y = 52. Find the y-intercept.
Step 1: Find slope: m = Δy/Δx = 6/2 = 3
Step 2: Use y = mx + b with m = 3 and (x, y) = (10, 52).
52 = 3*10 + b → b = 22 (y-intercept).
Step 3: plug the values for m & b into the equation: y = 3x + 22