Problem 43 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=4-x^{2}, \quad x \geq 0\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt{4 - x}\).

Step 1: Replace f(x) with y

Start by writing the equation in terms of y. Replace \(f(x)\) with \(y\) in the given function: \(y = 4 - x^2\). This makes it easier to solve for the inverse.

Step 2: Solve for x

We manipulate the equation to solve for \(x\). First, subtract 4 from both sides: \(y - 4 = -x^2\). Then, multiply both sides by -1 to eliminate the negative: \(-y + 4 = x^2\), or \(x^2 = 4 - y\).

Step 3: Take the square root

Since the value of \(x\) is non-negative, we only consider the positive root. Take the square root of both sides: \(x = \sqrt{4 - y}\).

Step 4: Swap x and y

To find the inverse, we swap \(x\) and \(y\). So, the inverse function is obtained: \(y = \sqrt{4 - x}\).

Step 5: Write the inverse function

The inverse function of \(f(x)\) is denoted by \(f^{-1}(x)\). Therefore, \(f^{-1}(x) = \sqrt{4 - x}\).

Key Concepts

Quadratic Functions

Quadratic functions are a key topic in algebra. They are expressed in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The defining feature here is the presence of the squared term, \(x^2\), which creates a parabolic graph. These graphs are symmetric and can open upwards or downwards depending on the sign of \(a\). For instance, in our original function \(f(x) = 4 - x^2\), we see a quadratic expression where the graph opens downwards, since the coefficient of \(x^2\) is negative.

Key properties of quadratic functions include:

• The vertex form, \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
• The axis of symmetry, which is a vertical line through the vertex, given by \(x = h\).
• The parabola opens upwards if \(a > 0\) and downward if \(a < 0\).

Understanding these features allows you to manipulate and invert such functions effectively.

Domain and Range

The domain and range of a function are fundamental concepts representing the input and output values a function can accept. The domain of a function refers to all possible 'x' values that you can substitute into the function, while the range refers to all possible 'y' values the function can produce.

For \(f(x) = 4 - x^2\) with \(x \geq 0\):

• The domain is \([0, \infty)\) because \(x\) must be greater than or equal to zero. This restriction is vital when finding the inverse, as it ensures that we work within the function's limits.
• The range for \(f(x)\) is \((-\infty, 4]\), because \(f(x)\) produces values from 4 down to negative infinity as \(x\) increases within its domain.

The concepts of domain and range are particularly critical when working with inverse functions. Properly identifying them ensures that the inverse function is both valid and accurate.

Square Root Functions

Square root functions belong to a category that involves the square root operation, typically expressed as \(f(x) = \sqrt{x}\). For these functions, it's crucial to consider domain restrictions to maintain the function's definition on real numbers. Generally, the expression inside the square root must be non-negative.

In the context of our inverse function, \(f^{-1}(x) = \sqrt{4 - x}\), we observe:

• The domain is \(x \leq 4\), because we cannot take the square root of a negative number and expect a real result. Thus, \(4 - x \geq 0\).
• The range is \([0, \, \infty)\), reflecting that the square root function outputs non-negative values.

When dealing with square root functions, always ensure you understand these domain and range limitations. This knowledge aids in graphing and ensures mathematical operations are valid.