A binary operation * is defined on a set of real numbers by x*y = x$^y$ for all values of x and y, if x * 2 = x, find the possible values of x.

Analysis

The problem defines a binary operation $x * y = x^y$. We are given the condition $x * 2 = x$ and asked to find the possible values of $x$ that satisfy this equation.

Concept

A binary operation is simply a rule for combining two numbers. In this case, the rule is to raise the first number to the power of the second number. To solve for $x$, we substitute the given values into the defined operation and solve the resulting algebraic equation.

Step-by-Step Solution

1. Apply the definition: Given $x * y = x^y$, substitute $y = 2$ into the expression: $x * 2 = x^2$

2. Set up the equation: We are told that $x * 2 = x$. Therefore: $x^2 = x$

3. Solve for $x$: Bring all terms to one side to form a quadratic equation: $x^2 - x = 0$

4. Factorize the equation: Factor out $x$: $x(x - 1) = 0$

5. Find the roots: For the product to be zero, either: $x = 0$ or $x - 1 = 0 \implies x = 1$

The possible values for $x$ are $0$ and $1$.

Conclusion

Comparing our result with the provided options, we see that $0$ and $1$ correspond to option C.

Exam Tip

When you have an equation like $x^2 = x$, never divide both sides by $x$ to get $x = 1$. If you do that, you lose the solution $x = 0$. Always move all terms to one side and factorize to ensure you capture all possible roots!