What is the range of f(x) = xยฒ for x โ {โ2, โ1, 0, 1, 2}? โ {0, 1, 4}.Outputs: (โ2)ยฒ=4, (โ1)ยฒ=1, 0ยฒ=0, 1ยฒ=1, 2ยฒ=4. Distinct values: {0, 1, 4}.
A function f has domain {1, 2, 3, 4}. Can it have a range of {5}? โ Yes โ all inputs can map to the same output.A constant function maps every input to the same output. The range {5} is valid.
A graph of a function is reflected over the y-axis. Is the result still a function? โ Yes โ a reflection over y-axis passes the vertical line test if the original does.Reflecting a function over the y-axis creates a new function. The vertical line test still passes because each x maps to exactly one y (just mirrored).
Which real-world scenario does NOT represent a function? โ A person โ list of their phone numbers.One person can have multiple phone numbers (one input, multiple outputs) โ not a function. The others have single outputs per input.
Function P has slope 3 and passes through (0, โ2). Function Q has slope 2 and passes through (0, 5). At what x-value do they have the same output? โ x = 7.P: y = 3x โ 2. Q: y = 2x + 5. Set equal: 3x โ 2 = 2x + 5 โ x = 7.
Compare functions: f given by y = โx + 8 and g given by table x: 0,2,4 | y: 6,4,2. Do they have the same slope? โ Yes โ both have slope โ1.f: m = โ1. g: slope = (4โ6)/(2โ0) = โ1. They have the same slope.
If f(0) = 5 and f(4) = 5, what can you say about the function? โ The y-intercept and f(4) are equal, but the function need not be constant.Two equal output values do not mean the function is constant โ many non-constant functions can produce the same output at two different inputs (e.g., y = xยฒ โ 4x + 5 gives 5 at x=0 and x=4).
Two delivery services: A charges $10 + $2 per mile; B charges $5 + $3 per mile. At what mileage are they equal in cost? โ 5 miles.10 + 2m = 5 + 3m โ 5 = m. At 5 miles the cost is the same ($20).
If f(2) = 8 and the function is linear with slope 3, what is f(0)? โ 2.f(0) = f(2) โ 2 ร slope = 8 โ 2(3) = 8 โ 6 = 2. (Move back 2 units at slope 3.)
Which of these statements is always true when comparing two linear functions? โ The function with greater slope increases faster for positive x.A greater slope means a faster rate of increase. For sufficiently large x, a larger-slope linear function will exceed any smaller-slope linear function.
A function doubles for every unit increase in x (y = 2หฃ). Between x = 5 and x = 6, how much does y increase? โ 32.y(5) = 32, y(6) = 64. ฮy = 32. The rate of change is increasing (nonlinear behavior).
Which real-world situation is modeled by a LINEAR function? โ A car traveling at constant speed (distance vs. time).Constant speed: d = rt means d is a linear function of t. Distance increases by the same amount each time unit.
A linear function passes through (1, 5) and (3, 11). What is the y-intercept? โ 3.Slope = (11โ5)/(3โ1) = 6/2 = 3. Using (1, 5): 5 = 3(1) + b โ b = 2. Wait: b = 5 โ 3 = 2. The y-intercept is 2.
The area of a square is A = sยฒ. Is this a linear or nonlinear function of s? โ Nonlinear โ s is squared (rate of change varies).A = sยฒ is quadratic. As s increases by 1, A increases by more each time. The rate of change is not constant.
Which property must hold for a function to be classified as linear? โ f(a + b) = f(a) + f(b) (additivity).A linear function satisfies superposition: f(a+b) = f(a) + f(b) and f(kx) = kf(x). This is the mathematical definition of linearity.
A table shows x = 0, 1, 2, 3 and y = 10, 7, 4, 1. Write the equation. โ y = โ3x + 10.Slope = โ3 (y decreases by 3 for each unit increase in x). y-intercept = 10 (when x = 0, y = 10). Equation: y = โ3x + 10.
Which is true of a quadratic function compared to a linear function? โ Quadratic's rate of change increases (for positive coefficients).For y = xยฒ, the slope increases as x increases (it's 2x at each point). This increasing rate of change is what makes it nonlinear.