Let the graph of t be a reflection in the y-axis followed by a translation 2 units left
of the graph of f(x) = x. Write a rule for t.

To find the rule for the function \( t(x) \) given that it's derived from \( f(x) = x \) through a reflection in the y-axis followed by a translation 2 units to the left, we can follow these steps:

### Step 1: Reflection in the y-axis
Reflecting \( f(x) = x \) in the y-axis changes the sign of the input variable \( x \). Therefore, the reflected function \( g(x) \) will be:
\[
g(x) = f(-x) = -x
\]

### Step 2: Translation 2 units to the left
Next, translating the function \( g(x) = -x \) 2 units to the left means replacing \( x \) with \( x + 2 \). The function \( t(x) \) will then be:
\[
t(x) = g(x + 2) = -(x + 2) = -x - 2
\]

### Final Answer:
The rule for the function \( t(x) \) is:
\[
t(x) = -x - 2
\]

Would you like more details or have any questions?

Here are some related questions you might consider:
1. What would be the rule if the translation was 2 units to the right instead of left?
2. How would the rule change if the reflection was in the x-axis instead of the y-axis?
3. What is the geometric interpretation of reflecting a function across the origin?
4. How would you graph the function \( t(x) = -x - 2 \)?
5. What would the rule be if you first translated and then reflected the function?

**Tip:** When working with transformations, always apply them in the correct order as specified in the problem.

Explore how to transform the function f(x) = x through a reflection in the y-axis followed by a translation 2 units to the left. Learn the rule for the transformed function t(x) = -x - 2 with detailed explanations and examples. Suitable for high school students learning about function transformations.

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