Domains, Composition, Inverses, and Graphs

A function is a rule with an agreement about which inputs are allowed. We will follow inputs through several rules, reverse a rule when possible, and interpret the same relationships on a graph.

What you will learn

• Determine valid inputs
• Compose and invert simple functions
• Interpret graphs

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

A rule and its allowed inputs

A function assigns exactly one output to each input in its domain. The domain is part of the function: the same expression on different input sets can describe different functions. A real square root requires a nonnegative radicand, while a denominator must be nonzero. A piecewise rule chooses its formula by testing the input against a stated condition; it does not ask you to choose whichever formula is convenient.

Function notation is a request to apply the rule. In $f(a+1)$, replace every occurrence of the input variable by the entire expression $a+1$, using parentheses where needed. Changing the output is different: $f(a)+1$ adds one after the rule has already been applied. Keeping those operations separate prevents many errors.

Composition follows an ordered path

The composition $f(g(x))$ means first apply $g$, then send its output into $f$. The starting value must lie in the domain of $g$, and the intermediate output must lie in the domain of $f$. Checking only the original input is insufficient. The order usually matters because the two rules change the input in different ways.

For a graph, a point $(a,b)$ states that input $a$ produces output $b$. Replacing the rule by $f(x-h)+k$ shifts that point to $(a+h,b+k)$. The input shift has the seemingly reversed sign because the new input must be $a+h$ to make the inner expression equal the old input $a$.

An inverse reverses an unambiguous rule

An inverse function reverses input and output. It exists on the chosen range only when no two allowed inputs produce the same output. To find a formula, write $y=f(x)$ and solve for $x$ in terms of $y$. The old output is now the input to the inverse. This process is not taking the reciprocal of the function value.

A linear rule with nonzero slope has a unique reverse. A squaring rule on all real numbers does not, because opposite inputs have the same square. Restricting its domain to nonnegative inputs makes the square-root function its inverse. Check an inverse by composing the two rules and recovering the original allowed input.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Confusing a composition with multiplication.
• Ignoring a restriction on the intermediate output.
• Reading $f^{-1}$ as $1/f$.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• A function includes its domain.
• Apply composed rules from inside outward.
• An inverse reverses a one-to-one input-output relationship.