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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 15
Chapter 3, Problem 15

The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = 2x + 3

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Start with the given function: \(f(x) = 2x + 3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = 2x + 3\).
Next, swap the roles of \(x\) and \(y\) to find the inverse: \(x = 2y + 3\). This means we are solving for \(y\) in terms of \(x\).
Isolate \(y\) by subtracting 3 from both sides: \(x - 3 = 2y\). Then, divide both sides by 2 to solve for \(y\): \(y = \frac{x - 3}{2}\).
Rewrite \(y\) as the inverse function notation: \(f^{-1}(x) = \frac{x - 3}{2}\). This is the formula for the inverse function.
To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f(x)\) and simplify to check if it equals \(x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}(x)\) and simplify to check if it equals \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

One-to-One Functions

A one-to-one function assigns each input a unique output and vice versa, ensuring that the function has an inverse. This property is essential because only one-to-one functions can be inverted, meaning each output corresponds to exactly one input.
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Decomposition of Functions

Finding the Inverse Function

To find the inverse of a function, swap the roles of x and y in the equation and solve for y. This process reverses the original function's operations, allowing you to express the inverse function f⁻¹(x) that 'undoes' f(x).
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Graphing Logarithmic Functions

Verification of Inverse Functions

Verifying an inverse involves showing that composing the function and its inverse returns the original input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms that the two functions are true inverses, effectively reversing each other's effects.
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Related Practice
Textbook Question

Use the graph of y = f(x) to graph each function g.

g(x) = −ƒ( x/2) +1

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Textbook Question

Find the average rate of change of the function from x1 to x2. f(x) = x² + 2x from x1 = 3 to x2 = 5

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Textbook Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope = -5, passing through (-4, -2)

Textbook Question

In Exercises 1-16, use the graph of y = f(x) to graph each function g.

g(x) = -f(2x) - 1

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Textbook Question

Find the domain of each function. f(x) = 1/[4/(x - 1) - 2]

Textbook Question

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (3√3, √5) and (−√3, 4√5)

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