Problem 53 - Algebra 2

Question

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=(x-4)^{2} $$

Step-by-Step Solution

Answer

The inverse of the function is given by \(f^{-1}(x) = \pm\sqrt{x}+4\), but \(f^{-1}\) is not a function as it does not satisfy the definition of a function. The domain of \(f(x)\) is \(-\infty < x < \infty\) while the range is \(0 \leq f(x) < \infty\). For the inverse, the domain is \(x \geq 0\) and range is all real numbers.

Step 1: Finding the inverse

To find the inverse of a function, one processes the original function in the reverse order. Given the original function \(f(x)=(x-4)^2\), the inverse \(f^{-1}(x)\) would be computed by replacing \(f(x)\) with \(y\), swapping \(x\) and \(y\), and then solving for \(y\). This gives \(x=(y-4)^2\). Taking the square root of both sides to solve for \(y\) would lead to two possible solutions, that is \(y = \pm\sqrt{x}+4\). This is because, by definition, \(y^2=x\) has two solutions for \(y\), which are \(+\sqrt{x}\) and \(-\sqrt{x}\).

Step 2: Determining the domain and range of the function

The domain of a function is the complete set of possible values of the independent variable, i.e., the input. For the function \(f(x)=(x-4)^2\), the domain is all real values of \(x\), represented as \(-\infty < x < \infty\). The range, on the other hand, is the complete set of possible values of the dependent variable, i.e., the output. Here, the lowest value of \(f(x)\) is 0, when \(x=4\). Hence, for \(f(x)\), the range is all numbers greater than or equal to zero, represented as \(0 \leq f(x) < \infty\).

Step 3: Determining the domain and range of the inverse

Based on the solution for the inverse, \(f^{-1}(x) = \pm\sqrt{x}+4\), it can be noted that the domain of \(f^{-1}(x)\) is \(x \geq 0\), because we can't get the square root of a negative number. However, since the inverse equation has a plus-minus symbol, it means for every value of \(x\) in the domain, there are two possible values for \(f^{-1}(x)\). Hence, the range of \(f^{-1}(x)\) will be all real number.

Step 4: Determining whether the inverse is a function

A relation is a function if for each input, there is exactly one output. This condition isn't satisfied by \(f^{-1}(x)\) since there are two possible values for \(f^{-1}(x)\) for any \(x\). Hence, \(f^{-1}(x)\) is not a function.

Key Concepts

Domain of a Function

Understanding the domain of a function is essential when dealing with any mathematical operation involving functions. The domain is essentially the set of all possible input values (usually represented by the variable \(x\)) that the function can accept. For example, in the quadratic function \(f(x) = (x-4)^2\), the domain is all real numbers. This is because there are no constraints or limitations when plugging values into \(x\).

For functions involving square roots, like inverse functions, the domain is often restricted to maintain valid real number outputs. Take, for instance, the inverse of our function, where the inverse relates to \(x = (y-4)^2\). Solving this gives \(y = \pm \sqrt{x} + 4\), thus the inputs \(x\) must be greater than or equal to zero to maintain real outputs, establishing \(x \geq 0\) as the domain.

• The function \(f(x) = (x-4)^2\) has a domain of all real numbers (-∞, ∞).
• The inverse \(f^{-1}(x) = \pm \sqrt{x} + 4\) has a domain of \(x \geq 0\).

Range of a Function

The range of a function is the complete set of all possible output values (often represented by \(f(x)\) or \(y\)). It's defined by what the function can produce as outputs based on the input domain. For \(f(x) = (x-4)^2\), the smallest output is 0, occurring when \(x = 4\). Therefore, the range of \(f\) is all non-negative real numbers, written as \([0, \infty)\). The inverse function \(f^{-1}(x) = \pm \sqrt{x} + 4\) suggests each \(x \geq 0\) maps to two possible \(y\) values (both a positive and negative square root), minus 4. This situation allows our inverse function to cover all real numbers, which means \(y\) can theoretically take any real value. Hence, the range for the inverse has no restrictions.

• The function \(f(x) = (x-4)^2\) has a range of \([0, \infty)\).
• The inverse \(f^{-1}(x)\) has a range of all real numbers (-∞, ∞).

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically in the form \(f(x) = ax^2 + bx + c\). A familiar example, like \(f(x) = (x-4)^2\), can be expanded to fit the standard form, revealing its structure as \(f(x) = x^2 - 8x + 16\). Quadratics have unique properties, such as being symmetrical about a vertical axis and having a distinct vertex, which in this case is a minimum point.

The graph of a quadratic is a parabola. Depending on the lead coefficient \(a\), the parabola can open upwards (when \(a > 0\)) or downwards (when \(a < 0\)). For our function \(f(x) = (x-4)^2\), the parabola opens upwards and the vertex or minimum point is at \((4, 0)\). Recognizing the vertex allows us to determine key characteristics like the minimum value, affecting the range.

• Quadratic functions are of the form \(ax^2 + bx + c\).
• The parabola's direction is determined by the sign of \(a\) (upwards if \(a > 0\)).
• The vertex of \(f(x) = (x-4)^2\) is at \((4,0)\), with its axis of symmetry at \(x=4\).