AP Precalculus: Parent Functions

To find an inverse by algebra: (1) write y = f(x), (2) swap x and y, (3) solve for y, and (4) name the result f⁻¹(x). To verify, show that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. If a function is not one-to-one, restrict its domain first and keep the sign that matches the restriction. For a rational function, after swapping you need to clear the fraction, collect all terms containing y on one side, and factor y out.

1. Parent Functions, Piecewise Functions, and Function Notation

Every function in this unit is built from a small family of parent functions. Learn their shapes, domains, and ranges and the rest is shifting and stretching.

• Constant y = c: domain all reals, range {c}
• Linear y = x: domain and range all reals
• Quadratic y = x²: domain all reals, range y ≥ 0
• Cubic y = x³: domain and range all reals
• Absolute value y = |x|: domain all reals, range y ≥ 0
• Square root y = √x: domain x ≥ 0, range y ≥ 0
• Reciprocal y = 1/x: domain and range all reals except 0

A piecewise function uses different rules on different parts of the domain. To evaluate it, first decide which interval the input belongs to, then use only that rule. Function notation works the same way: f(a) means replace every x with a, and f(x + h) means replace every x with the whole expression x + h. The difference quotient [f(x + h) − f(x)]/h measures an average rate of change and is simplified by expanding, canceling, and then dividing by h.

2. Vertical and Horizontal Translations

A translation slides a graph without changing its shape. Adding a constant outside the function, y = f(x) + d, moves the graph up d units (down if d is negative). Adding a constant inside, y = f(x + c), moves the graph left c units, and f(x − c) moves it right c units. The horizontal direction feels backward, so think of it this way: the new graph needs input x = −c to do what the old graph did at 0. Every point (a, b) on y = f(x) lands at (a − c, b + d) on y = f(x + c) + d.

3. Dilations and Reflections

A vertical dilation y = a·f(x) multiplies every output by a: it stretches away from the x-axis when |a| > 1 and shrinks toward it when |a| < 1. If a is negative, the graph is also reflected over the x-axis. A horizontal dilation y = f(bx) divides every input by b: it compresses by a factor of 1/|b| when |b| > 1 and stretches by a factor of 1/|b| when |b| < 1. If b is negative, the graph is reflected over the y-axis. A point (a, c) on y = f(x) lands at (a/b, k·c) on y = k·f(bx).

4. Combining Transformations, Domain, and Range

Write a transformed function as y = a·f(b(x + c)) + d. To build the graph, work in the same order you would evaluate: start inside the parentheses (horizontal shift and dilation), then apply the multiplier a (vertical dilation and reflection), and finish with the +d (vertical shift). To find the domain and range of a transformed graph, take the parent’s domain and range and apply the same moves: horizontal moves change the domain, and vertical moves change the range. A reflection or negative multiplier can flip the range, for example from y ≥ 0 to y ≤ 0.

5. Function Models: Linear and Quadratic

Choosing a model means matching the story to a function type. A linear model fits a constant rate of change, such as a fixed fee plus a per-unit price. A quadratic model fits data that rise and fall symmetrically, or any area problem, and its vertex gives the maximum or minimum. Draw a diagram, define the variable, and restrict the domain to inputs that make sense in the situation. For a linear model from a table, check that the rate of change is constant, then write y − y₁ = m(x − x₁). For a quadratic y = ax² + bx + c, the vertex is at x = −b/(2a). With data and a calculator, a regression model finds the best-fit line or parabola.

6. Function Models: Piecewise, Polynomial, and Rational

A piecewise model fits a situation whose rule changes, such as tiered pricing or taxes. Identify the independent variable, write one rule for each interval, and be careful that the second rule only applies to the usage beyond the cutoff, which is x − cutoff. A polynomial model fits data with several zeros or turning points, and the zeros and their multiplicities suggest the factored form, such as d = k·t(t − a)² where a graph that only touches the axis at a signals an even multiplicity. A rational model fits inverse variation, y = k/x, where xy = k stays constant, so x₁y₁ = x₂y₂. Cost problems with a fixed area often produce rational functions such as 2x + A/x.

7. Composition of Functions

The composite function (f ∘ g)(x) = f(g(x)) feeds the output of g into f. Work from the inside out: evaluate g first, then use that result as the input to f. To find the composite as an expression, replace every x in f with the entire expression for g. Composition is not commutative: f(g(x)) and g(f(x)) are usually different. The one special case is the identity function I(x) = x, which satisfies f(I(x)) = I(f(x)) = f(x). With tables, look up the inner output, then look that value up as the next input. With graphs, read the y-value of the inner function and use it as the x-value on the outer graph.

8. Domain of a Composite and Decomposing Functions

The domain of f(g(x)) contains the inputs x that are in the domain of g and whose output g(x) is in the domain of f. So find the domain of g, then remove any x for which g(x) is a value that f cannot accept. Decomposing a function runs composition backward: spot the “inside” expression (what sits in a radical, a denominator, an absolute value, or an exponent) and call it g(x), then call the remaining operation f. A function can be decomposed in several valid ways, but the simplest is usually best.

9. Inverse Functions, One-to-One, and the Horizontal Line Test

The inverse function f⁻¹ undoes f. If f(a) = b, then f⁻¹(b) = a, so every ordered pair (a, b) of f becomes (b, a) for the inverse. The domain and range swap too. A function has an inverse that is also a function only if it is one-to-one: different inputs always give different outputs. Graphically, use the horizontal line test: if every horizontal line meets the graph at most once, f is one-to-one. The graph of f⁻¹ is the reflection of the graph of f over the line y = x. If a function is not one-to-one, you can restrict its domain to a piece that is, such as x ≥ 0 for y = x².

10. Finding Inverses Algebraically

To find an inverse by algebra: (1) write y = f(x), (2) swap x and y, (3) solve for y, and (4) name the result f⁻¹(x). To verify, show that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. If a function is not one-to-one, restrict its domain first and keep the sign that matches the restriction. For a rational function, after swapping you need to clear the fraction, collect all terms containing y on one side, and factor y out.

Common Mistakes to Avoid

• Shifting the wrong way horizontally. f(x + 3) moves the graph left 3, and f(x − 3) moves it right 3.
• Mixing up inside and outside changes. Constants and multipliers inside the parentheses change x-values (horizontal), and those outside change y-values (vertical).
• Forgetting that a negative multiplier reflects. −f(x) flips over the x-axis, while f(−x) flips over the y-axis.
• Evaluating a composition in the wrong order. f(g(x)) means do g first. Reversing it usually gives a different answer.
• Dropping parentheses when substituting. For f(x) = x² − 3x, f(−2) = (−2)² − 3(−2), not −2² − 3(2).
• Using the second rule for the whole input of a piecewise model. After a cutoff, only the amount beyond the cutoff gets the new rate: x − cutoff.
• Finding an inverse of a function that is not one-to-one. Check the horizontal line test, or restrict the domain, before swapping x and y.

Frequently Asked Questions