Problem 57 - Algebra for College Students

Question

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=x^{2}-4 \quad \text { for } x \geq 0 $$

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \sqrt{x+4} \), and both functions are symmetric about \( y = x \).

Step 1: Understanding the Function

The function given is \( f(x) = x^2 - 4 \). It is specified that \( x \geq 0 \), which implies we are only considering the right side of the parabola. This means the domain of the function is \( x \geq 0 \) and the range is \( y \geq -4 \).

Step 2: Determine the Inverse Function

To find the inverse function, we start by replacing \( f(x) \) with \( y \): \( y = x^2 - 4 \). Next, solve for \( x \) in terms of \( y \). We get \( x^2 = y + 4 \) and \( x = \sqrt{y+4} \). Since \( x \geq 0 \), we take the positive square root. Thus, the inverse function is \( f^{-1}(x) = \sqrt{x+4} \).

Step 3: Set the Domain and Range of the Inverse

The range of the original function \( f \), \( y \geq -4 \), becomes the domain of \( f^{-1} \), \( x \geq -4 \). The domain of \( f \), \( x \geq 0 \), becomes the range of \( f^{-1} \), \( y \geq 0 \).

Step 4: Graphing the Functions

Plot \( f(x) = x^2 - 4 \) on the axes for \( x \geq 0 \). This is a parabola opening upwards, starting at the point \( (0, -4) \). Then plot \( f^{-1}(x) = \sqrt{x+4} \), which is the top half of a sideways parabola (a half-parabola) starting at \( (-4, 0) \). The two graphs will be symmetric across the line \( y = x \).

Step 5: Verify the Symmetry

To ensure correctness, verify that the graphs of \( f \) and \( f^{-1} \) are symmetrical along the line \( y = x \). Points on one graph should correspond to mirrored points on the other graph across this line.

Key Concepts

Graphing Functions

When learning about inverse functions, graphing them alongside the original function is crucial. Graphing helps to visually understand the relationship between the two functions and their symmetry across the line \( y = x \).
To graph a function such as \( f(x) = x^2 - 4 \) for \( x \geq 0 \), follow these steps:

• Identify the starting point of the parabola, which here is \( (0, -4) \) due to the \( -4 \) in the equation.
• Draw a parabola opening upwards from this point because the coefficient of \( x^2 \) is positive.

Next, for the inverse function \( f^{-1}(x) = \sqrt{x+4} \):

• Its graph is a "half-parabola" starting from \( (-4,0) \).
• As you move to the right, the graph rises slowly, depicting the square root transformation of the x-values.

Finally, check your graphs: they should mirror each other across \( y = x \), ensuring that all corresponding points on \( f \) and \( f^{-1} \) reflect through this line.

Parabola

A parabola is a U-shaped curve that is symmetrical. In the equation \( f(x) = x^2 - 4 \), it represents the graph of a quadratic function.
Quadratic functions are polynomials of degree two, and their graph forms a parabola. Here's how to identify key features:

• The vertex, which is the lowest or highest point (here, the lowest at \( (0, -4) \)).
• It opens upwards if the \( x^2 \) term is positive. For \( x \geq 0 \), we only consider the right side.

When graphing, remember:

• The parabola's width and direction can change with the coefficient of \( x^2 \).
• The vertex can be found using its formula if not directly evident.

For inverse functions, such as \( f^{-1}(x) = \sqrt{x+4} \), you see a horizontally oriented half-parabola. This is important for visualizing how inverse functions manipulate the basic shape of their original parabolas.

Domain and Range

The domain and range are fundamental concepts in understanding both functions and their inverses.
The domain of a function is the set of all possible input values (x-values). For \( f(x) = x^2 - 4 \), where \( x \geq 0 \), the domain is all non-negative numbers.
Meanwhile, the function's range is the set of all possible output values (y-values), here \( y \geq -4 \), due to the \( -4 \) hindering any values below -4.

• For the inverse \( f^{-1}(x) = \sqrt{x+4} \), the domain switches to \( x \geq -4 \), because values under -4 yield negative radicands, which are outside the real number scope in this context.
• And the range of the inverse, previously \( x \geq 0 \) in \( f(x) \), is \( y \geq 0 \), highlighting that the inverse evaluates only positive outputs.

Swapping domain and range between a function and its inverse helps in verifying correctness and understanding their nature more deeply. Make sure both the domain and range are always clearly defined for accurate graphing and solving purposes.