One to One Functions | Definition & Tests

Let’s explore Natural Exponential Functions! 🌿📈

The natural exponential function is a special type of exponential function where the base is the mathematical constant e. This e is approximately 2.71828. Because e is greater than 1 (e > 1), the natural exponential function behaves like the “growth” type of exponential function we discussed earlier.

Definition and Key Characteristics of f(x) = e^x 📊

The natural exponential function is defined as f(x) = e^x. It shares many fundamental graphical properties with other exponential functions of the form f(x) = a^x where a > 1:

• Domain: The domain of f(x) = e^x is all real numbers (ℝ) 🌍, meaning x can be any value from negative infinity to positive infinity.
• Range: The range of f(x) = e^x is all positive real numbers, excluding zero ((0, ∞)) 🎯. This means the output f(x) will always be greater than 0; the graph never touches or crosses the x-axis.
• Y-intercept: The graph of f(x) = e^x always crosses the y-axis at the point (0, 1) 📍. This is because e^0 = 1.
• X-intercept: There is no x-intercept 🚫. The graph approaches the x-axis but never reaches it.
• Monotonicity: Since its base e is greater than 1, f(x) = e^x is an increasing function 🌱📈. As x increases, f(x) also increases.
• One-to-One Function: Because it is a strictly increasing function, f(x) = e^x is a one-to-one function 🤝. This means it passes the Horizontal Line Test and has an inverse function (the natural logarithmic function).
• End Behaviour:
  • As x approaches positive infinity (x → ∞), f(x) also approaches positive infinity (e^x → ∞) ⬆️.
  • As x approaches negative infinity (x → -∞), f(x) approaches zero (e^x → 0) ➡️0. This means the x-axis acts as a horizontal asymptote.
• Special Slope and Area Property: The slope of the tangent line to f(x) = e^x at the point (1, e) is e. Additionally, the area under f(x) = e^x from (-∞, 1) is e.

Graphing f(x) = e^x with Ease 🖼️

Imagine a curve that starts very close to the negative x-axis on the left, crosses the y-axis at (0, 1), and then rapidly shoots upwards to the right.

       Y ^
         |
     6.0 +           /
         |          /
     4.0 +         /
         |        /
     2.0 +       /
         |      /
     1.0 +-----*----- (0,1)
         |    /
       0 +---+---------------> X
         -3 -2 -1 0 1 2 3

This visual representation aligns with the properties: it’s always above the x-axis (range is (0, ∞)), it passes through (0, 1), and it continually increases.

Practice Questions with Solutions 📝

Question 1: Consider the function f(x) = e^x. Which of the following statements about its graph are correct? 🤔 (a) The domain of f(x) is (0, ∞). (b) The graph of f(x) has an x-intercept. (c) f(x) is a decreasing function. (d) As x → -∞, f(x) approaches 0.

Solution:

Let’s evaluate each statement based on the characteristics of f(x) = e^x:

(a) The domain of f(x) is (0, ∞). * False ❌. The domain of the natural exponential function f(x) = e^x is all real numbers (ℝ).

(b) The graph of f(x) has an x-intercept. * False ❌. Exponential functions of the form f(x) = a^x (including e^x) do not have an x-intercept; they approach the x-axis but never touch or cross it.

(c) f(x) is a decreasing function. * False ❌. Since the base e is approximately 2.718 (which is > 1), f(x) = e^x is an increasing function 🌱📈.

(d) As x → -∞, f(x) approaches 0. * True ✅. For exponential functions with a base a > 1 (like e^x), as x approaches negative infinity, the function value approaches zero.

Therefore, only statement (d) is correct.

Question 2: A new social media platform’s user growth can be modelled by the function U(t) = 1000 * e^(0.1t), where U(t) is the number of active users after t months. (a) How many users did the platform start with (at t = 0)? 🚀 (b) Approximately how many users will there be after 5 months? (Use e^0.5 ≈ 1.649) 📈

Solution:

This question involves evaluating a transformed exponential function, similar to an example in the sources.

(a) Users at t = 0 (initial users): To find the number of users at the start, we substitute t = 0 into the function: U(0) = 1000 * e^(0.1 * 0) U(0) = 1000 * e^0 Since any non-zero number raised to the power of 0 is 1 (e^0 = 1): U(0) = 1000 * 1 U(0) = 1000 The platform started with 1,000 users 🚀.

(b) Users after 5 months: To find the number of users after 5 months, we substitute t = 5 into the function: U(5) = 1000 * e^(0.1 * 5) U(5) = 1000 * e^0.5 Given e^0.5 ≈ 1.649: U(5) ≈ 1000 * 1.649 U(5) ≈ 1649 There will be approximately 1,649 users after 5 months 📈.

Question 3: Describe the relationship between the graphs of f(x) = e^x and g(x) = e^(-x). Specifically, comment on their monotonicity and end behaviour as x → ∞. ↔️

Solution:

Let’s analyze both functions:

• f(x) = e^x:

  • Base: e ≈ 2.718 which is > 1.
  • Monotonicity: This is an increasing function 🌱📈.
  • End Behaviour as x → ∞: As x gets very large, e^x grows without bound, so f(x) → ∞ ⬆️.

• g(x) = e^(-x):

  • We can rewrite e^(-x) as (e^-1)^x or (1/e)^x.
  • Base: 1/e ≈ 1/2.718 ≈ 0.368. Since 0 < 0.368 < 1.
  • Monotonicity: This is a decreasing function 🍂📉.
  • End Behaviour as x → ∞: As x gets very large, (1/e)^x approaches 0, so g(x) → 0 ➡️0.

Relationship and Comparison:

• Monotonicity: f(x) = e^x is an increasing function, while g(x) = e^(-x) is a decreasing function.
• End Behaviour as x → ∞: As x approaches positive infinity, f(x) goes to infinity, whereas g(x) approaches zero.
• Graphical Transformation: Graphically, g(x) = e^(-x) is a reflection of f(x) = e^x across the y-axis.