To determine if the function $f(x) = 2 - 3x^2$ is injective, we need to check whether different inputs $x_1$ and $x_2$ can yield the same output $f(x_1) = f(x_2)$.

1. Assume $f(x_1) = f(x_2)$: $2 - 3x_1^2 = 2 - 3x_2^2$ By simplifying, we find: $-3x_1^2 = -3x_2^2$ Dividing both sides by -3 gives: $x_1^2 = x_2^2$

2. Consequences of $x_1^2 = x_2^2$: This implies: $x_1 = x_2 \quad \text{or} \quad x_1 = -x_2$ Thus, $f$ can produce the same output for two different inputs when $x_1$ and $x_2$ are negatives of each other.

Since we have found that $f(x_1) = f(x_2)$ does not necessarily imply $x_1 = x_2$ (e.g., $f(1) = f(-1) = -1$), the function is not injective.

Summary

The function $f(x) = 2 - 3x^2$ is not injective because it maps both $x$ and $-x$ to the same output.

Would you like more details or do you have any questions? Here are some related questions to consider:

1. What is the range of the function $f(x)$?
2. How can we determine if a function is surjective?
3. What are examples of injective functions?
4. How does the graph of $f(x)$ support the conclusion about injectivity?
5. What changes would make $f(x)$ injective?

Tip: Visualizing the graph of a function can often help in understanding its injectivity and other properties.