Question

Suppose a function \(f(x)\) has an inverse function, \(f^{-1}(x)\). a) Determine \(f^{-1}(5)\) if \(f(17)=5\). b) Determine \(f(-2)\) if \(f^{-1}(\sqrt{3})=-2\). c) Determine the value of \(a\) if \(f^{-1}(a)=1\) and \(f(x)=2 x^{2}+5 x+3, x \geq-1.25\).

Short answer

\(f^{-1}(5) = 17\), \(f(-2) = \sqrt{3}\), and \(a = 10\).

Step-by-step solution

Understanding the Inverse Function Property

The inverse function, denoted as \(f^{-1}(x)\), reverses the effect of the original function \(f(x)\). This means that if \(y = f(x)\), then \(x = f^{-1}(y)\).

Determine \(f^{-1}(5)\) given \(f(17) = 5\)

Since \(f(17) = 5\), by the property of inverse functions, \(f^{-1}(5) = 17\).

Determine \(f(-2)\) given \(f^{-1}(\sqrt{3}) = -2\)

Given \(f^{-1}(\sqrt{3}) = -2\), by the property of inverse functions, this means \(f(-2) = \sqrt{3}\).

Determine value of \(a\) such that \(f^{-1}(a) = 1\)

Given \(f(x) = 2x^2 + 5x + 3\), we need to find \(a\) such that \(f^{-1}(a) = 1\). This implies \(f(1) = a\). Calculate \(f(1)\): \[ f(1) = 2(1)^2 + 5(1) + 3 = 2 + 5 + 3 = 10. \] Therefore, \(a = 10\).

Key concepts

Inverse Function Property

When dealing with inverse functions, the core idea is that they reverse the operation of the original function. If the original function is denoted as \(f(x)\), then the inverse function is represented as \(f^{-1}(x)\). The key relationship here is that if \(y = f(x)\), then \(x = f^{-1}(y)\). This property helps us solve many problems where we need to find values of the inverse function.

For example, if we know that \(f(17) = 5\), by applying the inverse function property, we can see that \(f^{-1}(5) = 17\). This relationship always holds true and is fundamental to understanding and working with inverse functions effectively.

Determining Inverse Function Values

Once you grasp the inverse function property, you can determine specific values of the inverse function. Let's look at some practical examples.

Given the following problems:

• Find \(f^{-1}(5)\) if \(f(17) = 5\).
• Find \(f(-2)\) if \(f^{-1}(\text{√}3) = -2\).

By the inverse function property:

• Since \(f(17) = 5\), it means \(f^{-1}(5) = 17\).
• Since \(f^{-1}(\text{√}3) = -2\), it implies \(f(-2) = \text{√}3\).

As long as you remember the relationship, it's straightforward to determine the needed values.

Precalculus Problems

In precalculus, solving for inverses often involves dealing with more complex functions. Let's tackle an example using the function \(f(x) = 2x^2 + 5x + 3\) with \( x \geq -1.25\).

We need to find the value of \(a\) such that \(f^{-1}(a) = 1\). According to the inverse function property, this means \(f(1) = a\).

Calculate \(f(1)\): \[f(1) = 2(1)^2 + 5(1) + 3 = 2 + 5 + 3 = 10.\] Therefore, \(a = 10\).

This demonstrates how we can use the given function to find specific values and reinforces the importance of understanding the relationship between a function and its inverse.