a. Find the truth value of the biconditional statement of p and q where p represents '2 is an even number' and q represents '4 is an even number'. b. If f: R R be defined by f(x) = 2x 3 , find f^ 1…

a. Truth Value of the Biconditional Statement

A biconditional statement $p\leftrightarrow q$ is true only when both $p$ and $q$ have the same truth value (both true or both false).

Given:

• $p$: "2 is an even number."
• $q$: "4 is an even number."

Step 1: Determine the truth values of $p$ and $q$.

• $p$ is true because 2 is divisible by 2.
• $q$ is true because 4 is divisible by 2.

Step 2: Evaluate the biconditional statement. Since both $p$ and $q$ are true, the biconditional statement $p\leftrightarrow q$ is true.

Final Answer: The truth value of the biconditional statement $p\leftrightarrow q$ is true.

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b. Finding the Inverse Function $f^{-1}$

Given the function: $$f:\mathbb{R}\to \mathbb{R},f(x)=2x-3$$

Step 1: Replace $f(x)$ with $y$. $$y=2x-3$$

Step 2: Swap $x$ and $y$ to find the inverse. $$x=2y-3$$

Step 3: Solve for $y$. $$x+3=2y$$ $$y=\frac{x+3}{2}$$

Thus, the inverse function is: $$f^{-1}(x)=\frac{x+3}{2}$$

Final Answer: $$f^{-1}(x)=\boxed{\frac{x+3}{2}}$$

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c. Testing Even/Odd Nature and Symmetry of $y=8x^2$

1. Testing Even or Odd Nature

A function $f(x)$ is:

• Even if $f(-x)=f(x)$ for all $x$.
• Odd if $f(-x)=-f(x)$ for all $x$.

Given: $$y=f(x)=8x^2$$

Step 1: Compute $f(-x)$. $$f(-x)=8(-x{)}^2=8x^2=f(x)$$

Since $f(-x)=f(x)$, the function is even.

2. Testing Symmetry

A function can be symmetric about:

• The y-axis (even function).
• The origin (odd function).
• Neither.

Since $f(-x)=f(x)$, the graph of $y=8x^2$ is symmetric about the y-axis.

Conclusion

• Even/Odd Nature: The function is even.
• Symmetry: The graph is symmetric about the y-axis.

Final Answer: The function $y=8x^2$ is even and symmetric about the y-axis.