Q. Find the Inverse of the Function \( f(x) = 2x – 4 \)

Answer

Set up the equation.
Let y = 2x – 4.
Swap variables.
Swap x and y to find the inverse.

\[ x = 2y – 4 \]
Solve for y.
Isolate y.

\[ y = \frac{x + 4}{2} \]
State the final inverse function.
\[ f^{-1}(x) = \frac{x + 4}{2} \]

Detailed Explanation

Find the inverse of the function f(x) = 2x – 4 — step-by-step

Introduce a temporary variable for the output.
To solve for the inverse, set the function equal to y. This makes it easier to manipulate the equation algebraically.

\[ y = 2x – 4 \]

Explanation: We will solve this equation for x in terms of y so that we can reverse the roles of input and output.
Undo the operations on x in reverse order.
The right-hand side applies two operations to x: first multiply by 2, then subtract 4. To isolate x, perform the inverse operations in the reverse order: first add 4 to both sides, then divide both sides by 2.

Add 4 to both sides: \[ y + 4 = 2x \]

Divide both sides by 2: \[ x = \frac{y + 4}{2} \]

Explanation: Adding 4 cancels the −4. Dividing by 2 cancels the multiplication by 2. After these steps x is expressed in terms of y.
Swap variables to write the inverse as a function of x.
The inverse function takes the former output (y) as its new input. Replace y by f^{-1}(x) or, equivalently, swap x and y in the formula and call the result y again. Doing the swap gives:

\[ y = \frac{x + 4}{2} \]

Explanation: This expresses the inverse mapping: the input labeled x is returned to the original input of f.
Write the inverse function using standard notation.
Using the usual inverse-function notation, we have:

\[ f^{-1}(x) = \frac{x + 4}{2} \]

Explanation: This is the final formula for the inverse function.
Optional verification (composition).
Verify that composing f with its inverse returns the identity function on appropriate inputs.

Compute f(f^{-1}(x)): \[ f\bigl(f^{-1}(x)\bigr) = 2\left(\frac{x + 4}{2}\right) – 4 = (x + 4) – 4 = x. \]

Compute f^{-1}(f(x)): \[ f^{-1}\bigl(f(x)\bigr) = \frac{(2x – 4) + 4}{2} = \frac{2x}{2} = x. \]

Explanation: Both compositions give x, confirming the formulas are correct inverses of each other.
Domain and range remarks.
The original function f(x) = 2x – 4 is defined for all real numbers. Its inverse f^{-1}(x) = (x + 4)/2 is also defined for all real numbers. Therefore both domain and range are all real numbers.

Final answer: \[ f^{-1}(x) = \frac{x + 4}{2} \]

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Frequently Asked Questions

What is the inverse function of f(x) = 2x - 4?

f^-1(x) = (x + 4)/2. Swap x and y, solve for y, then rename y as f^-1(x).

How do you find the inverse step-by-step?

Replace f(x) with y: y = 2x - 4. Swap x and y: x = 2y - 4. Solve for y: y = (x + 4)/2. Rename y as f^-1(x).

Is f(x) = 2x - 4 invertible?

Yes. It's linear function with nonzero slope (2), so it's one-to-one and passes the horizontal line test; therefore it has an inverse on all real numbers.

How can I verify my inverse is correct?

Compose: f(f^-1(x)) = 2((x + 4)/2) - 4 = x and f^-1(f(x)) = (2x - 4 + 4)/2 = x. Both compositions give x, confirming the inverse.

What are the domain and range of f and f^-1?

For f(x) = 2x - 4, domain = (-∞, ∞) and range = (-∞, ∞). The inverse swaps domain and range, so f^-1 has the same domain and range (all real numbers).

How does the graph of the inverse relate to the original graph?

The graph of f^-1 is the reflection of the graph of f across the line y = x. For f(x) = 2x - 4, both are straight lines mirrored over y = x.

How do I find x such that f(x) = 10?

Use the inverse: x = f^-1(10) = (10 + 4)/2 = 7. So f(7) = 10.

Is the inverse also linear function?

Yes. The inverse of nonvertical linear function ax + b (with ≠ 0) is also linear. Here f^-1(x) = (x + 4)/2 is linear.

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