Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 12: Problem 3
Which point lies on the graph of \(f(x)=3^{x} ?\) A. (1,0) B. (3,1) C. (0,1) D. \(\left(\sqrt{3}, \frac{1}{3}\right)\)
Short Answer
Expert verified
The point (0, 1) lies on the graph of \(f(x) = 3^x\).
Step by step solution
01
- Understand the problem
We need to determine which point lies on the graph of the function \(f(x) = 3^x\). This means that the point \((a, b)\) must satisfy the equation \(b = 3^a\).
02
- Evaluate each option
We need to check each given point to see if it satisfies the equation \(b = 3^a\).
03
- Check point (1, 0)
For the point (1, 0): Substitute \(a = 1\) into \(f(x) = 3^x\). This gives \(f(1) = 3^1 = 3\). Since 3 does not equal 0, (1, 0) is not on the graph.
04
- Check point (3, 1)
For the point (3, 1): Substitute \(a = 3\) into \(f(x) = 3^x\). This gives \(f(3) = 3^3 = 27\). Since 27 does not equal 1, (3, 1) is not on the graph.
05
- Check point (0, 1)
For the point (0, 1): Substitute \(a = 0\) into \(f(x) = 3^x\). This gives \(f(0) = 3^0 = 1\). Since 1 equals 1, (0, 1) is on the graph.
06
- Check point \(\left(\sqrt{3}, \frac{1}{3}\right)\)
For the point \(\left(\sqrt{3}, \frac{1}{3}\right)\): Substitute \(a = \sqrt{3}\) into \(f(x) = 3^x\). This gives \(f(\sqrt{3}) = 3^{\sqrt{3}}\). Since \(3^{\sqrt{3}}\) is not equal to \(\frac{1}{3}\), \(\left(\sqrt{3}, \frac{1}{3}\right)\) is not on the graph.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Functions
Exponential functions are a type of mathematical function that can be written in the form: \[f(x) = a^x\] where 'a' is a positive constant and 'x' is a real number. These functions are unique because their growth rate is proportional to the value of the function. This means they can grow very quickly. A common example is the function \(f(x) = 3^x\), which we are working with in our exercise. Understanding exponential functions is essential in both algebra and calculus, as they arise frequently in real-world applications such as population growth, radioactive decay, and interest calculations in finance.
Function Evaluation
To evaluate a function means to find the value of the function at a particular input. In our exercise, this means substituting a given 'x' value into our function \(f(x) = 3^x\) and calculating the result. Here's a quick guide to evaluating functions:
- Identify the input value (the 'x' value).
- Substitute this value into the function.
- Simplify the expression to find the output value (the 'y' value or \(f(x)\)).
Graphing Points
Graphing points on an exponential function graph involves plotting specific coordinate points (x, y) on a Cartesian plane to see if they fit the exponential trend. In our problem, we determined if each given point satisfies the equation \(b = 3^a\).Here's a simple way to graph points on an exponential function:
- Choose input values for 'x'.
- Use the function to find corresponding 'y' values.
- Plot these (x, y) pairs on the graph.
Math Problem-Solving
Solving math problems efficiently involves a step-by-step approach and verifying each step thoroughly. In our exercise, we followed structured steps to determine which point lies on the graph of \(f(x) = 3^x\). Taking the right approach involves:
- Clearly understanding the problem.
- Breaking it into smaller, manageable steps.
- Evaluating each step systematically.
- Double-checking calculations to ensure accuracy.
- For (1, 0): \[f(1) = 3^1 = 3 \] which does not match 0.
- For (3, 1): \[f(3) = 3^3 = 27 \] which does not match 1.
- For (0, 1): \[f(0) = 3^0 = 1 \] which matches 1 perfectly.
- For \( \left(\sqrt{3}, \frac{1}{3}\right\): \[f(\sqrt{3}) = 3^{\sqrt{3}}\] which does not match \frac{1}{3}.
