Let's analyze the given function and the answers: Given: \( f(x) = 2x - 1 \) ### (a) Find the Inverse of \( f \) To find the inverse, solve for \( x \) in terms of \( y \): \[ y = 2x - 1 \] \[ y + 1 = 2x \] \[ x = \frac{y + 1}{2} \] So, the inverse function is: \[ f^{-1}(x) = \frac{x + 1}{2} \] The answer given in the box in the image is: \[ f^{-1}(x) = \frac{x}{2} + \frac{1}{2} \] which is algebraically equivalent to our answer. --- ### (b) Find the Domain of \( f \) \( f(x) = 2x - 1 \) is a linear function, so its domain is all real numbers. Looking at the answer options: - **Option A:** \( \{x \mid x \leq (-\infty, \infty)\} \) (this is just all real numbers) - **Option D:** "The domain is the set of all real numbers." Both A and D mean all real numbers, but D is the clearest way to state it. **Best Answer:** - **D. The domain is the set of all real numbers.** --- ### (c) Graphing You'd graph: - \( f(x) = 2x - 1 \) (a straight line, slope 2, y-intercept -1) - \( f^{-1}(x) = \frac{x + 1}{2} \) (a straight line, slope 1/2, y-intercept 1/2) - \( y = x \) (the identity line, slope 1, y-intercept 0) --- **Summary of answers:** - (a) \( f^{-1}(x) = \frac{x+1}{2} \) - (b) The domain is all real numbers (**Option D** is best) - (c) See the above explanation for graphing.

Let's carefully go through **Step 1: Finding the inverse of \(f(x) = 2x - 1\)**. **Step 1: Replace \(f(x)\) with \(y\):** \[ y = 2x - 1 \] **Step 2: Swap the variables \(x\) and \(y\):** \[ x = 2y - 1 \] **Step 3: Solve for \(y\) in terms of \(x\):** \[ x + 1 = 2y \] \[ y = \frac{x + 1}{2} \] **Step 4: Write the inverse function:** \[ f^{-1}(x) = \frac{x + 1}{2} \] **Summary:** To find the inverse, we switch \(x\) and \(y\) in the original function and then solve for \(y\). The inverse function is \(f^{-1}(x) = \frac{x + 1}{2}\). **Note:** This inverse function "undoes" the original function \(f(x)\). For example, if \(f(3) = 2(3) - 1 = 5\), then \(f^{-1}(5) = \frac{5 + 1}{2} = 3\), confirming they are inverse functions. --- Let me know if you'd like a visual explanation or further details!

Let's verify the options for the domain of \(f(x) = 2x - 1\): 1. **Option A:** \(\{x \mid x \leq (-\infty, \infty)\}\) - This notation is somewhat confusing. Typically, \(\leq\) is used for inequalities, but here it seems to imply the set of all \(x\) less than or equal to some value, but the value isn't specified. If interpreted as "all \(x\) such that \(x \leq \infty\)," it essentially includes all real numbers, since every real number is less than or equal to infinity. 2. **Option B:** \(\{x \mid x = []\}\) - Empty set; no elements. Incorrect, since the domain of a linear function is all real numbers. 3. **Option C:** \(\{x \mid |\text{some condition}|\}\) - The precise condition isn't specified here, but likely incomplete or incorrect. 4. **Option D:** "The domain is the set of all real numbers." - Correct for \(f(x) = 2x - 1\), which is defined for all real \(x\). **Conclusion:** The most accurate and clear choice is **Option D**: *The domain is the set of all real numbers.* --- **Summary:** | Option | Description | Correct? | |---------|----------------------------------------------------------|----------------------------| | A | The set of \(x\) such that \(x \leq \infty\) (interpreted as all real numbers) | Yes, but notation is unclear | | B | Empty set | No | | C | Incomplete or unclear condition | No | | D | The set of all real numbers | Yes, best choice | --- **Final note:** Since \(f(x) = 2x - 1\) is a linear function with no restrictions on \(x\), the domain is **all real numbers**, making **Option D** the best answer.