Problem 34
Question
For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=(x-4)^{2}, x \geq 4$$
Step-by-Step Solution
Verified Answer
The inverse function is \( f^{-1}(x) = \sqrt{x} + 4 \).
1Step 1: Understand the Function
The given function is \( f(x) = (x-4)^2 \) with the condition \( x \geq 4 \). Since the domain of \( f(x) \) is restricted to \( x \geq 4 \), \( f(x) \) is a one-to-one function and an inverse function can exist.
2Step 2: Replace f(x) with y
Replace \( f(x) \) with \( y \) to make calculations easier: \( y = (x-4)^2 \).
3Step 3: Swap Variables to Find Inverse
To find the inverse, swap \( x \) and \( y \): \( x = (y-4)^2 \).
4Step 4: Solve for y
Solve the equation \( x = (y-4)^2 \) for \( y \). Start by taking the square root of both sides: \( \sqrt{x} = |y-4| \). Considering \( y \geq 4 \), the equation simplifies to \( y - 4 = \sqrt{x} \).
5Step 5: Isolate y
Add 4 to both sides of the equation: \( y = \sqrt{x} + 4 \). Thus, the inverse function is \( f^{-1}(x) = \sqrt{x} + 4 \) with the domain \( x \geq 0 \).
6Step 6: Graph the Functions
Graph the function \( f(x) = (x-4)^2 \) defined for \( x \geq 4 \). It is a parabola opening upwards, starting at the point (4,0). Graph the inverse \( f^{-1}(x) = \sqrt{x} + 4 \), which is a square root function shifted 4 units upwards, starting at (0,4). The graph of the function and its inverse should be symmetric with respect to the line \( y = x \).
Key Concepts
Graphing FunctionsDomain and RangeOne-to-One Functions
Graphing Functions
Graphing functions is crucial when trying to understand the behavior and properties of mathematical expressions. It helps us visualize the relationship between variables and can quickly show us the range, intercepts, and overall structure of the graph.
For the function \( f(x) = (x-4)^2 \), the graph is a parabola which opens upwards. Starting from the vertex, the graph depicts the minimum point of the function.
The vertex form we're using, \( (x-4)^2 \), tells us the parabola shifts horizontally to the right by 4 units, starting at the point \( (4, 0) \).
When graphing the inverse function, \( f^{-1}(x) = \sqrt{x} + 4 \), the pattern is different. This function is a transformation of the basic square root function. It starts at \( (0, 4) \) and moves upwards as x increases.
When both functions are graphed, symmetrically placed about the line \( y=x \), they beautifully illustrate the inverse relationship. This line of symmetry represents each function transforming into the other when reflected across it.
For the function \( f(x) = (x-4)^2 \), the graph is a parabola which opens upwards. Starting from the vertex, the graph depicts the minimum point of the function.
The vertex form we're using, \( (x-4)^2 \), tells us the parabola shifts horizontally to the right by 4 units, starting at the point \( (4, 0) \).
When graphing the inverse function, \( f^{-1}(x) = \sqrt{x} + 4 \), the pattern is different. This function is a transformation of the basic square root function. It starts at \( (0, 4) \) and moves upwards as x increases.
When both functions are graphed, symmetrically placed about the line \( y=x \), they beautifully illustrate the inverse relationship. This line of symmetry represents each function transforming into the other when reflected across it.
Domain and Range
Understanding the domain and range is key in both creating and interpreting functions. It helps us define the set of possible input (domain) and output (range) values a function can have.
For the function \( f(x) = (x-4)^2 \), the domain is \( x \geq 4 \). This restriction means we only consider x-values equal to or greater than 4, preventing the parabola from mirroring to the left.
The range of this function starts at the lowest point of the parabola, \( y \geq 0 \), since squares are never negative.
Conversely, the inverse function \( f^{-1}(x) = \sqrt{x} + 4 \) has its domain \( x \geq 0 \), as we're dealing with square roots, which cannot be evaluated for negative numbers in the realm of real numbers.
Its range begins at 4, moving upwards infinitely as the values of x increase, \( y \geq 4 \). Understanding these restrictions helps in graphing and verifies that an inverse is possible.
For the function \( f(x) = (x-4)^2 \), the domain is \( x \geq 4 \). This restriction means we only consider x-values equal to or greater than 4, preventing the parabola from mirroring to the left.
The range of this function starts at the lowest point of the parabola, \( y \geq 0 \), since squares are never negative.
Conversely, the inverse function \( f^{-1}(x) = \sqrt{x} + 4 \) has its domain \( x \geq 0 \), as we're dealing with square roots, which cannot be evaluated for negative numbers in the realm of real numbers.
Its range begins at 4, moving upwards infinitely as the values of x increase, \( y \geq 4 \). Understanding these restrictions helps in graphing and verifies that an inverse is possible.
One-to-One Functions
A one-to-one function is one where each x-value corresponds to exactly one y-value, and each y-value corresponds to exactly one x-value. This characteristic is essential for a function to have an inverse that is also a function.
The function \( f(x) = (x-4)^2 \) is made one-to-one by restricting its domain to \( x \geq 4 \). This restriction ensures that the parabola only moves in the direction where each y-value corresponds back to only one x-value.
Without such a restriction, the function wouldn't pass the horizontal line test, a quick visual check that each horizontal line crosses the graph at most once, indicating a one-to-one function.
By ensuring that both the function and its inverse are one-to-one, we allow them to pair symmetrically across the line \( y=x \). It highlights not only how functions invert but how restricting domains can preserve their functional properties.
The function \( f(x) = (x-4)^2 \) is made one-to-one by restricting its domain to \( x \geq 4 \). This restriction ensures that the parabola only moves in the direction where each y-value corresponds back to only one x-value.
Without such a restriction, the function wouldn't pass the horizontal line test, a quick visual check that each horizontal line crosses the graph at most once, indicating a one-to-one function.
By ensuring that both the function and its inverse are one-to-one, we allow them to pair symmetrically across the line \( y=x \). It highlights not only how functions invert but how restricting domains can preserve their functional properties.
Other exercises in this chapter
Problem 33
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