What is the domain and range?
f(x)=3^x-4
To find the domain and range of the function f(x) = 3^x - 4, let's start by understanding what the domain and range represent.
The domain refers to all the possible input values (x-values) that the function can take. It represents the set of values for which the function is defined. In other words, it's the set of x-values that we can plug into the function to obtain a valid output.
The range, on the other hand, represents all the possible output values (y-values) that the function can produce. It is the set of values that the function can take on as its output.
To determine the domain, we need to consider any restrictions or conditions that the function may have. In this case, the function is f(x) = 3^x - 4. A key thing to note is that the base of the exponent is positive and the exponent itself can be any real number. Therefore, there are no restrictions or limitations on the value of x. Hence, the domain of this function is all real numbers, or (-∞, +∞).
To find the range, we can analyze how the function behaves. The base of the exponent is 3, which is positive, meaning that the function will never output a negative value. Additionally, as x increases, 3^x also increases exponentially. Since we subtract 4 from the exponential function, it will shift the graph down by 4 units.
This indicates that the range of the function f(x) = 3^x - 4 will be all real numbers less than or equal to -4, as the function's output will always be less than -4.
To summarize:
Domain: All real numbers (-∞, +∞)
Range: All real numbers less than or equal to -4 (-∞, -4]