Functions and Their Properties
What this domain tests
The Functions and their properties domain tests the function concept across algebraic and graphical representations: notation, domain and range, composition, inverses, transformations, and basic behavior of parent functions. CLEP College Algebra favors questions that combine two ideas, such as finding the domain of a composition.
Domain and range
- The domain is the set of allowed inputs. Exclude values that make a denominator zero and, for even roots, values that make the radicand negative.
- The range is the set of outputs. Use the graph, vertex, asymptotes, or end behavior to determine it.
- Express domains and ranges in interval notation, using parentheses at excluded values and infinities.
Composition and inverses
- $(f\circ g)(x)=f(g(x))$. Its domain requires $x$ in the domain of $g$ and $g(x)$ in the domain of $f$.
- An inverse exists only for a one-to-one function; the horizontal line test detects whether each output has a unique input.
- To find an inverse, swap $x$ and $y$ and solve for $y$. The domain and range of $f$ and $f^{-1}$ interchange, and $f(f^{-1}(x))=x$ on the appropriate domain.
Transformations and parent graphs
- $y=f(x-h)+k$ shifts the graph $h$ units right and $k$ units up.
- $y=af(x)$ stretches vertically when $|a|>1$ and compresses when $0<|a|<1$; negative $a$ reflects across the $x$-axis.
- $y=f(-x)$ reflects across the $y$-axis. Apply transformations in a consistent order, handling horizontal shifts and scalings inside the argument carefully.
Worked example
Find the inverse of $f(x)=\frac{2x+1}{x-3}$.
Write $y=\frac{2x+1}{x-3}$ and swap: $x=\frac{2y+1}{y-3}$. Multiply: $x(y-3)=2y+1$, so $xy-3x=2y+1$. Collect $y$ terms: $xy-2y=3x+1$, hence $y(x-2)=3x+1$ and $y=\frac{3x+1}{x-2}$. Thus $f^{-1}(x)=\frac{3x+1}{x-2}$, with $x\ne2$. The original function excludes $x=3$ from its domain, and its inverse excludes $x=2$ from its domain, matching the range restriction.
Traps that cost marks
- Assuming $f(g(x))=g(f(x))$; composition is generally not commutative.
- Ignoring the inner function's domain when finding the domain of a composition.
- Reading $f^{-1}(x)$ as $1/f(x)$ rather than as inverse notation.
- Confusing horizontal and vertical line tests, or shifting a graph in the wrong direction.