IMAT Mathematics: Part 2

An interactive quiz on Functions and Graphs.

Fundamental Concepts Review

Domain and Range: The domain is the set of all possible input values ($x$). The range is the set of all possible output values ($f(x)$). Be mindful of restrictions from square roots (argument $\ge 0$) and fractions (denominator $\neq 0$).

Inverse & Composite Functions: For an inverse $f^{-1}(x)$, the domain and range are swapped. For a composition $(f \circ g)(x)$, the domain depends on the domains of both $f$ and $g$.

Graph Transformations: For $y = a \cdot f(b(x-c)) + d$, apply transformations in order: horizontal stretch/reflection (b), horizontal shift (c), vertical stretch/reflection (a), and vertical shift (d). Always factor inside the function to find the true horizontal shift.

Problem 6: Composition of Functions and Domain

Hard
FunctionsCompositionDomain and Range

Given the functions $f(x) = \frac{1}{x-2}$ and $g(x) = \sqrt{x-1}$, what is the domain of the composite function $(f \circ g)(x)$?

A

$x \ge 1$

B

$x > 5$

C

$x \ge 1$ and $x \neq 5$

D

$x > 2$