One to One Functions | Definition & Tests

Let’s dive into Graphing Exponential Functions! 📈📉

In our previous discussion, we established that an exponential function is generally defined as f(x) = a^x, where the base a is a positive constant (a > 0) and not equal to one (a ≠ 1). Now, let’s explore how these functions look when plotted and what their key characteristics are.

Core Characteristics of Exponential Graphs (f(x) = a^x) 📊

Regardless of the specific value of the base a (as long as it meets the definition criteria), all standard exponential functions f(x) = a^x share some fundamental graphical properties:

• Domain: The domain of an exponential function f(x) = a^x is all real numbers (R). This means you can input any real number as the exponent x.
• Range: The range of f(x) = a^x is all positive real numbers, excluding zero ((0, ∞)). This implies that the output of an exponential function will always be positive, and the graph will never touch or cross the x-axis.
• Y-intercept: Every exponential function f(x) = a^x has a y-intercept at the point (0, 1). This is because any non-zero number raised to the power of zero equals one (e.g., a^0 = 1).
• X-intercept: There is no x-intercept for f(x) = a^x. The graph will approach the x-axis as x tends to negative infinity (when a > 1) or positive infinity (when 0 < a < 1), but it will never actually cross or touch it.

Behavior Based on the Base a 🧭

The graph of an exponential function f(x) = a^x behaves differently depending on whether the base a is greater than 1 or between 0 and 1.

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

When a > 1, the function exhibits exponential growth.

• Monotonicity: It is an increasing function. This means as x gets larger, the value of f(x) also gets larger at an accelerating rate. For instance, f(x) = 2^x is an increasing function, where x1 < x2 implies 2^x1 < 2^x2.

• End Behaviour:

  • As x approaches positive infinity (x → ∞), f(x) also approaches positive infinity (a^x → ∞).
  • As x approaches negative infinity (x → -∞), f(x) approaches zero (a^x → 0). This is why the x-axis acts as a horizontal asymptote.

• Example Graph: f(x) = 2^x

  Y ^
    |
  30|              /
  20|             /
  10|            /
    |           /
    |          /
    |         /
   1+--------*-------*-----> X
    |       (0,1)
    |

  (This is a simplified representation; actual curve grows much faster)

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

When 0 < a < 1, the function exhibits exponential decay.

• Monotonicity: It is a decreasing function. This means as x gets larger, the value of f(x) gets smaller, approaching zero. For example, f(x) = (1/2)^x is a decreasing function, where x1 < x2 implies (1/2)^x1 > (1/2)^x2.

• End Behaviour:

  • As x approaches positive infinity (x → ∞), f(x) approaches zero (a^x → 0). Again, the x-axis is a horizontal asymptote.
  • As x approaches negative infinity (x → -∞), f(x) approaches positive infinity (a^x → ∞).

• Example Graph: f(x) = (1/2)^x

  Y ^
    |     /
  30|    /
  20|   /
  10|  /
    | /
   1+*-------*-------*-----> X
    (0,1)
    |

  (Again, a simplified representation)

The Natural Exponential Function (f(x) = e^x) 🌿

A particularly significant exponential function is the natural exponential function, where the base is the mathematical constant e (approximately 2.718).

• It is defined as f(x) = e^x.
• Since e > 1, the natural exponential function behaves like the a > 1 case: it is an increasing function.
• Its domain is R and range is (0, ∞).
• It passes through (0, 1) [Recall from general characteristics].
• The slope of the tangent line to f(x) = e^x at the point (1, e) is e.
• The area under f(x) = e^x from (-∞, 1) is e.

Connection to One-to-One Functions 🤝

As we’ve discussed, a function is one-to-one if each unique input maps to a unique output [Lec 30 - Polynomials]. This can be visually checked using the Horizontal Line Test. If any horizontal line intersects the graph of a function at most once, then the function is one-to-one.

Exponential functions are one-to-one functions. This is because they are strictly increasing (a > 1) or strictly decreasing (0 < a < 1) across their entire domain. This property ensures that they will always pass the Horizontal Line Test. Consequently, because they are one-to-one, exponential functions possess inverse functions, which are logarithmic functions.

Practice Questions with Solutions 📝

Question 1: Consider the function h(x) = (0.3)^x. Which of the following statements about its graph are true? 🤔 (a) The function has an x-intercept at (0, 0). (b) The function is an increasing function. (c) The range of the function is (0, ∞). (d) As x → -∞, h(x) → 0.

Solution:

Let’s analyze h(x) = (0.3)^x based on the characteristics of exponential functions:

• Base a: Here, a = 0.3. Since 0 < 0.3 < 1, this is an exponential decay function.

(a) The function has an x-intercept at (0, 0). * False. Exponential functions of the form f(x) = a^x never have an x-intercept; they approach the x-axis but never cross it. The y-intercept is always (0, 1).

(b) The function is an increasing function. * False. Because the base a = 0.3 is between 0 and 1, h(x) is a decreasing function. As x increases, h(x) decreases.

(c) The range of the function is (0, ∞). * True. The range of any exponential function f(x) = a^x is all positive real numbers, (0, ∞). This means the output values are always greater than zero.

(d) As x → -∞, h(x) → 0. * False. For functions with a base 0 < a < 1, as x approaches negative infinity (x → -∞), h(x) approaches positive infinity (a^x → ∞). The graph goes upwards to the left.

Therefore, only statement (c) is true. ✅

Question 2: Compare the graphs of f(x) = 4^x and g(x) = (1/4)^x. Which statement accurately describes their relationship? 🤔 (a) Both functions are decreasing. (b) Both functions have the same end behavior as x → ∞. (c) Both functions are one-to-one. (d) The y-intercept of f(x) is (4, 0).

Solution:

Let’s examine each function:

• f(x) = 4^x: Base a = 4. Since a > 1, f(x) is an increasing exponential function.
• g(x) = (1/4)^x: Base a = 1/4. Since 0 < a < 1, g(x) is a decreasing exponential function.

Now let’s evaluate the statements:

(a) Both functions are decreasing. * False. f(x) = 4^x is increasing, while g(x) = (1/4)^x is decreasing.

(b) Both functions have the same end behavior as x → ∞. * False. For f(x) = 4^x (base a > 1), as x → ∞, f(x) → ∞. For g(x) = (1/4)^x (base 0 < a < 1), as x → ∞, g(x) → 0. Their end behaviors are opposite.

(c) Both functions are one-to-one. * True. Both f(x) and g(x) are strictly monotonic (one increasing, one decreasing). Functions that are increasing or decreasing are one-to-one. This means they both pass the Horizontal Line Test.

(d) The y-intercept of f(x) is (4, 0). * False. For any exponential function f(x) = a^x, the y-intercept is always (0, 1).

Therefore, only statement (c) is accurate. ✅

Question 3: An initial investment P grows exponentially over time t (in years) according to the formula A(t) = P * (1.05)^t. Which statement best describes the growth observed in this function’s graph? 🤔 (a) The function shows exponential decay. (b) The graph will eventually cross the x-axis, indicating the investment reaches zero. (c) The function will continuously increase as time passes. (d) The initial investment P cannot be negative for the function to be valid.

Solution:

Let’s analyze the given function A(t) = P * (1.05)^t. This is a transformed exponential function, f(x) = P * a^x, where P is a scaling factor and a = 1.05 is the base. For simplicity in terms of graphing the base behavior, we can focus on a = 1.05.

• Base a: Here, a = 1.05. Since a > 1, the core exponential behavior is one of growth.

(a) The function shows exponential decay. * False. Since the base 1.05 is greater than 1, the function represents exponential growth, not decay.

(b) The graph will eventually cross the x-axis, indicating the investment reaches zero. * False. The range of a standard exponential function a^x is (0, ∞). Even with a positive scaling factor P, the function will never reach or cross the x-axis; it will always remain positive. An investment cannot become literally zero or negative with this type of growth model.

(c) The function will continuously increase as time passes. * True. Because the base 1.05 is greater than 1, the function (1.05)^t is an increasing function. Assuming P is a positive initial investment, the total amount A(t) will therefore continuously increase over time.

(d) The initial investment P cannot be negative for the function to be valid. * Outside Source Knowledge: The sources provided define exponential functions and discuss their properties, but they do not explicitly state rules about the sign of the constant P in a real-world application context like this. However, in the context of investment, P (initial principal) is conventionally non-negative. If P were negative, it would represent a debt, and its absolute value might grow or shrink depending on the context, but it wouldn’t fit the typical “investment growth” scenario. Therefore, while generally true in the real-world context of investment, this specific constraint is not directly found in the provided mathematical sources on exponential functions.

Therefore, statement (c) best describes the growth observed. ✅