One to One Functions | Definition & Tests

Let’s explore Exponential Functions in an easy-to-understand way! 🚀

What is an Exponential Function? 🤔

An exponential function is a mathematical function that shows rapid growth or decay. It’s defined with a constant base raised to a variable exponent.

Formally, an exponential function in standard form is described as: f(x) = a^x

Where:

• a is the base.
• a must be greater than 0 (a > 0).
• a cannot be equal to 1 (a ≠ 1).
• x is the variable exponent.

Think of it like compound interest, where your money grows (or shrinks) at an accelerating rate! 💰📈📉

The Natural Exponential Function (Base e) 🌿

A very important type of exponential function is the natural exponential function, where the base is the mathematical constant e.

• It’s defined as f(x) = e^x.
• The value of e is approximately 2.718, so e > 1.
• Domain: All real numbers (R). This means you can put any real number as the exponent x.
• Range: All positive real numbers, excluding zero (0, ∞). This means the output f(x) will always be positive.
• Y-intercept: The point (0, 1). If you put x = 0, e^0 = 1.
• X-intercept: There is no x-intercept. The graph never touches or crosses the X-axis.
• End Behaviour:
  • As x approaches positive infinity (x → ∞), e^x also approaches positive infinity (e^x → ∞).
  • As x approaches negative infinity (x → -∞), e^x approaches 0 (e^x → 0).

General Characteristics of f(x) = a^x based on the Base a 📊

The behaviour of an exponential function largely depends on its base a.

1. When the Base a is Greater Than 1 (a > 1) 🌱

• Example: f(x) = 2^x or f(x) = 3^x.
• Domain: All real numbers (R).
• Range: All positive real numbers (0, ∞).
• Y-intercept: The point (0, 1).
• X-intercept: Nil.
• End Behaviour:
  • As x → ∞, a^x → ∞.
  • As x → -∞, a^x → 0.
• Monotonicity: It is an increasing function. This means as x gets larger, f(x) also gets larger.
• One-to-One Function: Because it’s a strictly increasing function, f(x) = a^x (for a > 1) is a one-to-one function. This property is crucial because it means these functions have inverse functions (logarithmic functions).

2. When the Base a is Between 0 and 1 (0 < a < 1) 🍂

• Example: f(x) = (1/2)^x.
• Domain: All real numbers (R).
• Range: All positive real numbers (0, ∞).
• Y-intercept: The point (0, 1).
• X-intercept: Nil.
• End Behaviour:
  • As x → ∞, a^x → 0.
  • As x → -∞, a^x → ∞.
• Monotonicity: It is a decreasing function. This means as x gets larger, f(x) gets smaller.
• One-to-One Function: Since it’s a strictly decreasing function, f(x) = a^x (for 0 < a < 1) is also a one-to-one function.

Laws of Exponents 🔢

Exponential functions follow specific rules for their exponents: For any real numbers s and t, and positive bases a and b:

• Product Rule: a^s × a^t = a^(s+t)
  • Example: 2^3 × 2^4 = 2^(3+4) = 2^7
• Power of a Power Rule: (a^s)^t = a^(s×t)
  • Example: (2^3)^4 = 2^(3×4) = 2^12
• Power of a Product Rule: (ab)^s = a^s b^s
  • Example: (2×3)^4 = 2^4 × 3^4

These laws are fundamental for manipulating and simplifying exponential expressions.

Connection to One-to-One Functions (Revisited) 🤝

As we discussed in our previous conversation, a function is one-to-one if each unique input maps to a unique output [Lec 30 - Polynomials]. This is crucial for inverse functions to exist [Lec 30 - Polynomials].

Exponential functions are one-to-one because they are either consistently increasing (a > 1) or consistently decreasing (0 < a < 1) across their entire domain. This property ensures that for any given output, there was only one possible input that could have produced it. Therefore, an inverse function (f⁻¹) always exists for an exponential function, and this inverse is the logarithmic function.

Practice Questions with Solutions 📝

Question 1: Which of the following expressions represents an exponential function? Justify your answer. 🤔 (a) f(x) = x^5 (b) g(x) = 7^x (c) h(x) = (-2)^x (d) k(x) = 1^x

Solution: Only (b) g(x) = 7^x represents an exponential function. ✅

• Justification:
  • An exponential function must be in the form f(x) = a^x, where the base a > 0 and a ≠ 1.
  • For g(x) = 7^x, the base a = 7. Since 7 > 0 and 7 ≠ 1, it fits the definition of an exponential function.
  • (a) f(x) = x^5 is a power function (a polynomial), not an exponential function, because the base x is variable, not a constant.
  • (c) h(x) = (-2)^x is not an exponential function because the base a = -2 is not greater than 0 (-2 < 0). This would lead to undefined real values for certain x (e.g., (-2)^(1/2)).
  • (d) k(x) = 1^x is not an exponential function because the base a = 1 is excluded from the definition (a ≠ 1). 1^x would always simply be 1, which is a constant function.

Question 2: Simplify the expression (5^2 × 5^3)^2 using the laws of exponents. 🔢

Solution: We can simplify the expression step-by-step using the laws of exponents:

1. Apply the Product Rule a^s × a^t = a^(s+t) for the terms inside the parenthesis: 5^2 × 5^3 = 5^(2+3) = 5^5
2. Apply the Power of a Power Rule (a^s)^t = a^(s×t): (5^5)^2 = 5^(5×2) = 5^10

Therefore, (5^2 × 5^3)^2 = 5^10. ✅

Question 3: Consider the function f(x) = (0.8)^x. (a) Is this function increasing or decreasing? (b) Is this function a one-to-one function? Justify your answer.

Solution:

(a) The function f(x) = (0.8)^x is a decreasing function. 📉

• Justification: The base a = 0.8. Since 0 < a < 1, the exponential function f(x) = a^x is a decreasing function. This means as the value of x increases, the value of f(x) decreases.

(b) Yes, the function f(x) = (0.8)^x is a one-to-one function. ✅

• Justification: According to the source, if a function is an increasing or decreasing function, then it is one-to-one. Since f(x) = (0.8)^x is a decreasing function, it inherently satisfies the condition of being one-to-one, meaning different inputs will always yield different outputs.

Question 4: A new smartphone’s battery life B(t) (in hours) after t years of use is modelled by the function B(t) = 20 * (0.9)^t. (a) What was the initial battery life of the smartphone (at t = 0)? (b) What will be the battery life after 5 years? (Round to two decimal places.)

Solution:

(a) To find the initial battery life, we set t = 0 [Outside source knowledge, but standard function evaluation]. B(0) = 20 * (0.9)^0 Since any non-zero number raised to the power of 0 is 1, (0.9)^0 = 1. B(0) = 20 * 1 = 20 The initial battery life of the smartphone was 20 hours. 🔋

(b) To find the battery life after 5 years, we set t = 5. B(5) = 20 * (0.9)^5 B(5) = 20 * 0.59049 [Calculation performed externally] B(5) = 11.8098 Rounding to two decimal places, the battery life after 5 years will be approximately 11.81 hours. 📉

I hope this comprehensive explanation of Exponential Functions, complete with emojis, examples, and practice questions, has made the concept much clearer for you! 😊