Math Problem Statement

Sketch a graph of each of the following: a. 𝑓𝑓(π‘₯π‘₯) = 𝑏𝑏 π‘₯π‘₯ if 𝑏𝑏 > 1 b. 𝑓𝑓(π‘₯π‘₯) = 𝑏𝑏 π‘₯π‘₯ if 0 < 𝑏𝑏 < 1 c. 𝑓𝑓(π‘₯π‘₯) = log𝑏𝑏(π‘₯π‘₯) if 𝑏𝑏 > 1 d. 𝑓𝑓(π‘₯π‘₯) = log𝑏𝑏(π‘₯π‘₯) if 0 < 𝑏𝑏 < 1

Solution

Here is the graph of the four functions you requested:

  1. Exponential function f(x)=bxf(x) = b^x where b>1b > 1 (increasing curve)
  2. Exponential function f(x)=bxf(x) = b^x where 0<b<10 < b < 1 (decreasing curve)
  3. Logarithmic function f(x)=log⁑b(x)f(x) = \log_b(x) where b>1b > 1 (increasing curve)
  4. Logarithmic function f(x)=log⁑b(x)f(x) = \log_b(x) where 0<b<10 < b < 1 (decreasing curve)

Let me know if you need further details or if you'd like to discuss any specific part of the graph!

Related questions:

  1. How does the value of bb affect the steepness of an exponential graph?
  2. What happens to the logarithmic graph as the base approaches 1?
  3. How do the exponential and logarithmic graphs relate to each other?
  4. How can you find the horizontal asymptotes of these functions?
  5. What are some real-world applications of exponential and logarithmic functions?

Tip:

Exponential and logarithmic functions are inverses of each other, meaning if you have y=bxy = b^x, then x=log⁑b(y)x = \log_b(y).

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Logarithmic Functions
Graphing Functions

Formulas

f(x) = b^x
f(x) = log_b(x)

Theorems

Exponential and Logarithmic Function Properties
Inverse Function Theorem

Suitable Grade Level

Grades 9-11

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