Math Problem Statement
Solution
Let's analyze each function to determine if they are one-to-one:
Definition:
A function is one-to-one (injective) if every element of the domain maps to a unique element in the codomain. In simpler terms, no two different inputs give the same output.
(A)
- The square root function is not one-to-one because it only gives non-negative outputs, and values like , meaning different inputs can give the same output.
- Not one-to-one.
(B)
- Exponential functions like (where ) are always one-to-one because they are strictly increasing, meaning each input will give distinct outputs.
- One-to-one.
(C)
- Logarithmic functions are the inverse of exponential functions, and they are also strictly increasing for , meaning different inputs give different outputs.
- One-to-one.
(D)
- This is a linear function with a non-zero slope. Linear functions with non-zero slopes are one-to-one because they are strictly increasing or decreasing.
- One-to-one.
(E)
- The function is quadratic, which is not one-to-one. A parabola opens upward or downward, and different inputs can yield the same output (e.g., ).
- Not one-to-one.
Final Answer:
The one-to-one functions are B, C, and D.
Would you like further details or have any questions?
Here are 5 related questions you might explore:
- How do we determine if a quadratic function is one-to-one?
- Can all exponential functions be classified as one-to-one?
- How do the properties of logarithmic functions relate to one-to-one criteria?
- Why does the square root function fail to be one-to-one?
- What role does the derivative play in determining whether a function is one-to-one?
Tip: A quick way to test if a function is one-to-one is using the horizontal line test: if any horizontal line crosses the graph more than once, the function is not one-to-one.
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Math Problem Analysis
Mathematical Concepts
One-to-One Functions
Exponential Functions
Logarithmic Functions
Quadratic Functions
Linear Functions
Formulas
f(x) = √(x - 1)
f(x) = 5.6^x
f(x) = log_3(x)
f(x) = 3x - 3
f(x) = 3x^2 + 5
Theorems
One-to-One Function Criterion
Horizontal Line Test
Suitable Grade Level
Grades 10-12
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