describe the domain, range,asymptotes and if the graph increase or decrease, and is this is a funtion from this equation y=5-3^x
To describe the domain, range, asymptotes, and whether the graph increases or decreases for the equation y = 5 - 3^x, we can break it down step by step:
Domain:
The domain of a function refers to the set of all possible input values (x-values) that the function can accept. In this case, since the base of the exponential term is 3^x, the domain includes all real numbers.
Range:
The range of a function represents the set of all possible output values (y-values) that the function can produce. For the given equation y = 5 - 3^x, the base of the exponential term, 3^x, will always be positive or zero, which means that the output values range from negative infinity up to 5 but never exceed it.
Asymptotes:
To determine the asymptotes, we need to consider the behavior of the equation as x approaches positive or negative infinity. Since the exponential term 3^x grows very rapidly either positively or negatively, we can conclude that there are no asymptotes as x approaches infinity or negative infinity. In other words, the graph has no horizontal asymptotes.
Increase or Decrease:
To examine whether the graph increases or decreases, we can analyze the behavior of the given equation. The term 3^x is an exponentially increasing function, while the function y = 5 - 3^x subtracts it from a constant value of 5. As a result, the overall function y = 5 - 3^x decreases as x increases. In other words, the graph of the equation will decrease as x moves to the right along the x-axis.
In summary:
- Domain: All real numbers
- Range: Values from negative infinity up to, but never exceeding, 5
- Asymptotes: No horizontal asymptotes
- Increase or Decrease: The graph decreases as x increases.