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We are asked to find the largest possible value of \( a \) such that the function \( f(x) = x^2 + 4x + 5 \) has an inverse when defined on \( x \leq a \).

### Step 1: Analyze the Function
The function \( f(x) = x^2 + 4x + 5 \) is a quadratic function, and quadratic functions are not one-to-one (i.e., they don't have inverses over their entire domain). However, we can restrict the domain so that the function becomes one-to-one.

### Step 2: Determine the Turning Point
For the quadratic function to be one-to-one, it must either be strictly increasing or strictly decreasing in the restricted domain. The turning point (vertex) of a parabola given by \( f(x) = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). For our function:

\[
x_{\text{vertex}} = -\frac{4}{2(1)} = -2
\]

Thus, the vertex is at \( x = -2 \). The function will be decreasing to the left of \( x = -2 \) and increasing to the right of \( x = -2 \).

### Step 3: Choose the Domain
To ensure that the function is one-to-one, we must restrict its domain to either the left or right of the vertex. Since we are asked to define \( f(x) \) on \( x \leq a \), we will take the domain \( x \leq -2 \) to ensure the function is one-to-one (strictly decreasing on this interval).

### Step 4: Largest Possible Value of \( a \)
The largest possible value of \( a \) is the \( x \)-coordinate of the vertex because, at this point, the function changes from decreasing to increasing. Therefore, the largest \( a \) such that \( f(x) \) has an inverse is:

\[
a = -2
\]

### Conclusion:
The largest possible value of \( a \) is \( \boxed{-2} \).

Would you like further details on any step or have any questions?

Here are five related questions to extend your understanding:
1. How do we determine if a function is one-to-one?
2. Can a quadratic function ever have an inverse over its entire domain?
3. What happens to the shape of a quadratic function when its leading coefficient is negative?
4. How would the solution change if we were asked to find the smallest possible value of \( a \)?
5. How can the concept of inverse functions be applied to real-world problems?

**Tip:** Always check the behavior of a function (increasing or decreasing) to determine if it can have an inverse over a certain interval.

Consider the function defined on x <= a by f(x) = x^2 + 4x + 5. Find the largest possible value of a if f has an inverse.

Solve a problem involving the quadratic function f(x) = x^2 + 4x + 5, where we need to find the largest value of 'a' such that f has an inverse. Key concepts include quadratic functions, vertex of a parabola, and one-to-one functions. Ideal for high school students (Grades 10-12).

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