Let's go over the problem in the photo.

The function in question is $g(x) = x^2 - 1$, and the task is to use the definition of a one-to-one function to determine whether it is one-to-one or not.

Step 1: Understanding the Definition of a One-to-One Function

To show a function $g(x)$ is one-to-one, we need to use the definition that states:
If $g(a) = g(b)$, then $a = b$.

This means that no two distinct inputs (a and b) should map to the same output. If two different inputs produce the same output, then the function is not one-to-one.

Now let's analyze $g(x) = x^2 - 1$:

To prove whether the function is one-to-one, assume that: $g(a) = g(b)$ This means: $a^2 - 1 = b^2 - 1$ Simplifying: $a^2 = b^2$ This implies: $a = b \quad \text{or} \quad a = -b$

Step 2: Conclusion

From this, we can see that $a = b$ is not the only solution. If $a = -b$, then $g(a) = g(b)$ even though $a \neq b$. This shows that two different inputs can produce the same output, which means the function is not one-to-one.

Part 1 of 5:

The correct completion for Part 1 is: To show a function $g$ is a one-to-one function using the definition, it must be shown that if $g(a) = g(b)$, then $a = b$.

Now, if you need help with Parts 2-5, let me know, and I can walk you through them as well!

Would you like to continue with the next parts?