QuestionIdentify which functions are one-to-one: , , , , .
Studdy Solution
STEP 1
Assumptions1. We are given five functions and we need to determine which of them are one-to-one. . A function is one-to-one (also known as injective) if every element of the function's domain maps to a unique element of its range. In other words, no two different elements in the domain of the function have the same image in the range of the function.
STEP 2
To check if a function is one-to-one, we can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.
STEP 3
Let's start by analyzing the first function .
STEP 4
This function can be simplified to .
STEP 5
The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.
STEP 6
Next, let's analyze the second function .
STEP 7
The graph of this function is a square root function shifted to the left by9 units and vertically stretched by a factor of5. This function is one-to-one because it passes the horizontal line test.
STEP 8
Next, let's analyze the third function .
STEP 9
The graph of this function is a hyperbola, which is not one-to-one because it fails the horizontal line test.
STEP 10
Next, let's analyze the fourth function .
STEP 11
The graph of this function is a cubic function, which is one-to-one because it passes the horizontal line test.
STEP 12
Finally, let's analyze the fifth function .
STEP 13
The graph of this function is a quartic function, which is not one-to-one because it fails the horizontal line test.
Therefore, the one-to-one functions are and .
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