Unit 1 · Topic 1.1

Change in Tandem

Learn to tell one consistent story about how a function's input and output change together, no matter whether the function is presented in words, a table, an equation, or a graph.

Learning Goals

Identify the domain, range, independent variable, and dependent variable of a function.
Find an image from an input and find every preimage associated with an output.
Use comparisons of function values to describe where a function is increasing or decreasing.
Translate among verbal, numerical, analytical, and graphical representations.
Construct and justify a graph from a contextual description, including zeros and concavity.

1. A Function Connects Two Quantities

A function assigns exactly one output to each permitted input. The input set is the domain; the set of resulting outputs is the range.

The input is the independent variable.
The output is the dependent variable because it depends on the input.
Two functions are equal only when they have the same domain and give the same output for every input in that domain.

\[x\longmapsto f(x)\]

2. Image and Preimage

The image of an input is its single function output. A preimage of an output is any input that produces that output, so one output can have several preimages.

For \(f(x)=x^2-4\), the image of 3 is

\[f(3)=3^2-4=5.\]

To find the preimages of 5, solve \(x^2-4=5\). The output 5 has two preimages: \(x=-3\) and \(x=3\).

3. Compare Inputs and Outputs Together

Choose two inputs in the domain and keep each input paired with its output. The changes are

\[\Delta x=b-a,\qquad \Delta y=f(b)-f(a).\]

Comparison	Input change	Output change	Conclusion
From \(x=-3\) to \(x=0\)	\(\Delta x=3\)	\(\Delta y=-4-5=-9\)	Input rises while output falls.
From \(x=0\) to \(x=3\)	\(\Delta x=3\)	\(\Delta y=5-(-4)=9\)	Input and output both rise.

The sign of \(\Delta y\) describes the direction of change over the selected pair. Topic 1.2 will turn these changes into a rate by dividing by \(\Delta x\).

4. Increasing and Decreasing

A function is increasing on an interval when every ordered pair of inputs \(a<b\) in that interval satisfies \(f(a)<f(b)\).

\[a<b\Longrightarrow f(a)<f(b)\]

It is decreasing when every such pair satisfies \(f(a)>f(b)\).

\[a<b\Longrightarrow f(a)>f(b)\]

For \(f(x)=x^2-4\), the function decreases to \(x=0\), reaches a minimum, and then increases. AP Precalculus does not assess choosing open versus closed endpoints when reporting these intervals.

5. Four Equivalent Representations

Verbal	Describe which quantity controls the other and how their directions or rates change.
Numerical	Read ordered pairs from a table and compare successive inputs and outputs.
Analytical	Use a formula such as \(f(x)=x^2-4\) to calculate and compare exact values.
Graphical	Read points, increasing or decreasing behavior, curvature, intercepts, and extrema.

A correct translation preserves the same domain, paired values, and behavior. A graph is not decoration; it is the complete set of input-output pairs.

6. Read the Graph's Story

The graph decreases toward its minimum at \((0,-4)\), then increases, crosses the x-axis at \(x=\pm2\), and is concave up throughout.

An x-intercept occurs when the output is zero. These inputs are the function's zeros.
Concave up means the rate of change is increasing; concave down means the rate of change is decreasing.
Increasing/decreasing describes output direction. Concavity describes how the rate itself changes. They are different ideas.

7. Build a Graph from Context

Suppose water depth in a widening tank increases as time passes, but each new minute adds less depth than the previous minute.

Put time on the horizontal axis and depth on the vertical axis.
The graph rises because depth increases.
The graph becomes less steep because depth rises more slowly.
Therefore the graph is increasing and concave down.

Always label axes and units, honor the contextual domain, and show only features supported by the description.

8. Worked AP-Style Example

A function \(P(t)=-2t^2+12t+5\) gives a quantity for \(0\le t\le6\). Compare its behavior before and after \(t=3\).

\[P(0)=5,\qquad P(3)=23,\qquad P(6)=5.\]

From \(t=0\) to \(t=3\), input and output increase together. At \(t=3\), the output reaches its maximum of 23. From \(t=3\) to \(t=6\), input increases while output decreases.

AP-ready conclusion: “On the first part of the contextual domain, \(P\) is increasing because larger inputs produce larger outputs. After \(t=3\), \(P\) is decreasing because larger inputs produce smaller outputs.”

9. Common Errors

Comparing outputs without stating the corresponding inputs or interval.
Calling a relation a function when one input is paired with multiple outputs.
Finding only one preimage when the equation has more than one valid solution.
Assuming two or three table rows prove behavior over every point of an interval.
Confusing “increasing but slowing down” with decreasing.
Ignoring domain restrictions or units in a contextual model.

Key Takeaways

Describe how a function's inputs and outputs vary together by comparing values and translating among verbal, numerical, analytical, and graphical representations.
Core relationship: \(a
Error check: Compare ordered input-output pairs and name the interval; a few isolated values alone do not prove behavior across an entire interval.

Checkpoint · Topic 1.1

Let \(g(x)=x^2-6x+5\) on \(0\le x\le6\).

Find \(g(0)\), \(g(3)\), and \(g(6)\).
Describe how the input and output vary together before and after \(x=3\).
Find the zeros and explain what they represent on the graph.
State whether the graph is concave up or concave down and justify your answer from its shape.