Example Question - function properties
Here are examples of questions we've helped users solve.
Name of the Function with Constant Value
<p>Функция, значение которой не изменяется и остается постоянным при всех значениях аргумента в своей области определения, называется константной функцией.</p> <p>Таким образом, \( y = f(x) \) - константная функция.</p>
Determining the Domain and Range from a Graph
<p>The domain of a function is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (y-values).</p> <p>Looking at the provided graph, it appears to be a straight line without any breaks or holes, which indicates that the line extends infinitely in both the positive and negative directions along the x-axis.</p> <p>This means the domain of the function is all real numbers.</p> <p>$$ Domain: (-\infty, \infty) $$</p> <p>Similarly, the line extends infinitely in both the positive and negative directions along the y-axis, which means the range of the function is also all real numbers.</p> <p>$$ Range: (-\infty, \infty) $$</p>
Graph Analysis of a Function
<p>\textbf{(a) The domain of } f:</p> <p>[\text{All real numbers}] \text{, since the graph extends infinitely in the x-direction.}</p> <p>\textbf{(b) The range of } f:</p> <p>[-3, \infty) \text{, because the highest y-value the graph reaches is infinite and the lowest is } -3.</p> <p>\textbf{(c) The zeros of } f:</p> <p> \{ -4, 2 \} \text{, the x-values where the graph intersects the x-axis.}</p> <p>\textbf{(d) } f(-3.5):</p> <p> \text{As } x = -3.5, \text{ f(x) is about } 2.5 \text{, reading from the graph.}</p> <p>\textbf{(e) The intervals on which } f \text{ is increasing:}</p> <p>(-\infty, -4) \cup (2, \infty) \text{, the intervals on the x-axis where the graph goes upwards as x increases.}</p> <p>\textbf{(f) The intervals on which } f \text{ is decreasing:}</p> <p>(-4, 2) \text{, the interval on the x-axis where the graph goes downwards as x increases.}</p> <p>\textbf{(g) The values for which } f(x) \leq 0:</p> <p>[-4, 2] \text{, these are the x-values where the graph is at or below the x-axis.}</p> <p>\textbf{(h) Any relative maxima or minima:}</p> <p>\text{Relative maximum at } (2, 3) \text{, relative minimum at } (-4, -3) \text{ based on the graph's high and low points respectively.}</p> <p>\textbf{(i) The value(s) of } x \text{ for which } f(x) = -3:</p> <p>\{-4, 0\} \text{, the x-values where the graph touches the horizontal line } y = -3.</p> <p>\textbf{(j) Is } f(0) \text{ positive or negative?}</p> <p>\text{Negative, since the point } (0, f(0)) \text{ lies below the x-axis where } f(0) \text{ is around } -3.</p>
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