Free Printable Inverse Functions Worksheets for Students - Free Printable

Let's go through the problem step by step based on the image you provided. The task involves:

1. Determining if given graphs are functions using the vertical line test.
2. Determining if functions are one-to-one using the horizontal line test.
3. Understanding what an inverse function is.

We’ll analyze each graph accordingly.

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🔹 Part 1: Determine if the graphs are functions (Vertical Line Test)

A function must pass the vertical line test: no vertical line should intersect the graph more than once.

#### Graph 1: Parabola opening upward (U-shaped)
- This is a standard parabola like $ y = x^2 $.
- A vertical line (e.g., at $ x = 1 $) intersects the graph only once.
- ✔ Passes vertical line test → It IS a function.

#### Graph 2: Horizontal "sideways" parabola (like $ x = y^2 $)
- This graph opens to the right and left.
- A vertical line (e.g., at $ x = 1 $) will intersect the graph twice (top and bottom).
- ✘ Fails vertical line test → NOT a function.

#### Graph 3: Straight diagonal line (like $ y = x $)
- A straight line with slope.
- Any vertical line intersects it exactly once.
- ✔ Passes vertical line test → It IS a function.

✔ Summary for Part 1:
- Graph 1: Function
- Graph 2: Not a function
- Graph 3: Function

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🔹 Part 2: Determine if the functions are One-to-One (Horizontal Line Test)

A function is one-to-one if each y-value corresponds to only one x-value, meaning no horizontal line intersects the graph more than once.

> Only one-to-one functions have inverses.

#### Graph 1: Parabola (U-shaped)
- A horizontal line (e.g., $ y = 1 $) intersects the graph twice (once on the left, once on the right).
- ✘ Fails horizontal line test → NOT one-to-one.

#### Graph 2: Cubic-like curve (increasing with inflection point)
- This looks like a cubic function like $ y = x^3 $.
- A horizontal line crosses it only once at any y-value.
- ✔ Passes horizontal line test → ONE-TO-ONE.

#### Graph 3: Diagonal straight line
- This is $ y = x $ or similar.
- Every horizontal line intersects it only once.
- ✔ Passes horizontal line test → ONE-TO-ONE.

✔ Summary for Part 2:
- Graph 1: Not one-to-one
- Graph 2: One-to-one
- Graph 3: One-to-one

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🔹 Part 3: What is an Inverse Function?

As stated:
- Only one-to-one functions have inverse functions.
- A function performs a "DO" operation; its inverse performs the "UNDO".
- Example: If $ f(x) = 2x $, then $ f^{-1}(x) = \frac{x}{2} $.

So, only the second and third graphs (the cubic and linear ones) have inverses because they are one-to-one.

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✔ Final Answers:

#### Are the graphs functions? (Vertical Line Test)
1. Parabola (U-shape): Yes
2. Sideways parabola: No
3. Diagonal line: Yes

#### Are the functions one-to-one? (Horizontal Line Test)
1. U-shaped parabola: No
2. Cubic-like curve: Yes
3. Diagonal line: Yes

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🧠 Key Takeaways:

| Graph | Function? | One-to-One? | Has Inverse? |
|-------|-----------|-------------|--------------|
| 1 | Yes | No | ✘ No |
| 2 | Yes | Yes | ✔ Yes |
| 3 | Yes | Yes | ✔ Yes |

> Note: Even though Graph 2 passes the vertical line test, it’s not a function in the first set — but in the second section, it appears again as a valid function. So we assume the second set shows valid functions that we're testing for one-to-oneness.

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💡 Summary Explanation:

- Vertical Line Test: Checks if a graph represents a function.
- Horizontal Line Test: Checks if a function is one-to-one.
- Only one-to-one functions have inverses.
- Inverse functions "undo" the original function.

So, among the graphs shown:
- Only the cubic and linear graphs have inverses.

Let me know if you'd like help sketching the inverses of those two!