Math Problem Statement

Which of the following are one-to-one functions? Select all that apply.

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) f(x)=x1f(x) = \sqrt{x - 1}

  • The square root function is not one-to-one because it only gives non-negative outputs, and values like f(2)=f(3)=1f(2) = f(3) = 1, meaning different inputs can give the same output.
  • Not one-to-one.

(B) f(x)=5.6xf(x) = 5.6^x

  • Exponential functions like axa^x (where a>1a > 1) are always one-to-one because they are strictly increasing, meaning each input x1x2x_1 \neq x_2 will give distinct outputs.
  • One-to-one.

(C) f(x)=log3xf(x) = \log_3{x}

  • Logarithmic functions are the inverse of exponential functions, and they are also strictly increasing for x>0x > 0, meaning different inputs give different outputs.
  • One-to-one.

(D) f(x)=3x3f(x) = 3x - 3

  • 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) f(x)=3x2+5f(x) = 3x^2 + 5

  • 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., f(1)=f(1)f(-1) = f(1)).
  • 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:

  1. How do we determine if a quadratic function is one-to-one?
  2. Can all exponential functions be classified as one-to-one?
  3. How do the properties of logarithmic functions relate to one-to-one criteria?
  4. Why does the square root function fail to be one-to-one?
  5. 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