Justify The Model (2.1, 2.2, 2.6)
A table shows outputs increasing by +5 each time x increases by 1.
What model fits, and why?
Linear
An arithmetic sequence has first term 10 and common difference −2.
Write the nth-term rule.
10 – 2n
In f(g(x)), what role does g(x) play?
g(x) provides the input for f
What does the Horizontal Line Test tell you?
It determines whether each output corresponds to exactly one input (invertibility).
“Exponential functions can change by the same amount.”
True or false — explain.
False. Exponential functions change by a constant factor, not a constant amount.
A quantity doubles every time x increases by 3.
Is the model linear or exponential? Explain in one sentence.
Exponential; Doubling means the outputs change by a constant factor not a constant difference.
A geometric sequence starts at 5 and triples each step.
Write the nth-term rule.
5(3)^n
Why does order matter when composing functions?
Because changing the order changes which outputs are used as inputs, which can change the result.
Why does an increasing function always have an inverse?
Because each output comes from exactly one input, so the inverse is also a function.
“A larger initial value always means larger outputs.”
Why is this sometimes false?
Because exponential growth with a smaller initial value can eventually surpass a linear model with a larger starting value.
Why does an exponential model eventually outperform a linear model with a larger initial value?
Exponential growth eventually outpaces linear growth because it multiplies repeatedly, while linear growth only adds a constant amount.
An exponential function has initial value 4 and growth factor 1.2.
Write the function.
f(x) = 4(1.2)^x
Which domain restriction matters more when finding the domain of f(g(x)): f(x) or g(x)?
The domain of g, because g is evaluated first.
What does reflecting over y=x represent conceptually?
It represents swapping inputs and outputs.
“Any function can have an inverse.”
What’s missing from this statement?
The function must be one-to-one (pass the Horizontal Line Test).
A situation shows rapid early growth that slows over time.
Which model is least appropriate: linear, exponential, or quadratic?
Exponential is least appropriate, because exponential change does not slow over time; it accelerates.
A population decreases by 15% each year starting at 200.
What is the base of the model?
The base is 0.85.
What must be true for f(g(x)) to be defined?
The output of g(x) must be in the domain of f(x).
Why must the domain of a quadratic be restricted before finding an inverse?
Because without restricting the domain, the function fails the Horizontal Line Test.
“Linear and exponential growth look the same on small intervals.”
Why can this be misleading?
Because exponential growth can look nearly linear over short intervals.
Why can both linear and exponential models sometimes appear to fit a small data set?
Over a small interval, exponential growth can appear almost linear, so both models may seem to fit initially.
A quantity doubles every 6 hours.
Write the model using doubling time.
f(t) = a(2)^(t/6)
Why does composition often create more domain restrictions than either function alone?
Because the output of one function becomes the input of another, adding additional restrictions.
Why does f(f^-1(x))=x but only for values in a certain domain?
Because the inverse only exists for values in the range of the original function.
“Regression always gives the best model.”
Why is context still important?
Because regression does not account for context, and the model must make sense for the situation.