Example Question - increasing and decreasing intervals
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Analysis of a Function Using Its Graph
<p>The domain of \( f \): All real numbers \( x \) such that \( -4 \leq x \leq 5 \).</p> <p>The range of \( f \): All real numbers \( y \) such that \( -3 \leq y \leq 5 \).</p> <p>The zeros of \( f \): \( x = -2, 1 \).</p> <p>\( f(-3.5) \): Cannot be determined exactly from the graph; not provided.</p> <p>The intervals on which \( f \) is increasing: \( (-4, -3) \cup (1, 5) \).</p> <p>The intervals on which \( f \) is decreasing: \( (-3, 1) \).</p> <p>The values for which \( f(x) < 0 \): \( x \) in intervals \( (-2, 1) \).</p> <p>Any relative maxima or minima: Relative maximum at \( x = -3 \), relative minimum at \( x = 1 \).</p> <p>The value(s) of \( x \) for which \( f(x) = 3 \): Approximately \( x = 4.5 \).</p> <p>Is \( f(0) \) positive or negative? Negative, since \( f(0) \) is below the \( x \)-axis.</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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