Determine whether the graph below is that of a function by u

Determine whether the graph below is that of a function by using the vertical-line test. If it is use the graph to find its domain and range. the intercepts if any symmetry with respect to the x-axis y-axis of the origin. (Type your answers in interval no talon.) The graph is not a function. What are the intercepts? Select the correct choice below and fill in any answer boxes within your choice. (Type an ordered pair. (Use a comma to separate answer as needed.) There are no intercepts the graph is not a function. Determine it the graph a symmetrical with respect to the x-axis y-axis or the origin.

Solution

The vertical line test helps in determining whether a curve is a graph of a function or not. A function of a single variable can only have one output, y, for each unique input, x. If a vertical line intersects a curve on the xy-plane more than once, then for one value of x the relation has more than one value of y, and so, the curve does not represent a function. If all vertical lines intersect a curve at most once then the curve represents a function.

(a) Here, any vertical line intersects the curve at most once, hence the curve represents a function.

(b) The y-intercept(where x = 0, i.e. where the curve meets the Y-Axis) is 6. The x-intercept(where y = 0, i.e. where the curve meets the X-Axis) is 1.

(c) There is no symmetry about either the X-Axis or the Y-Axis.The graph represents a parabola which opens upwards and has its vertex at (4,-9). Let its equation be y =a(x-4)²-9 where a is an arbitrary real number. Since the parabola passes through the point (0,6), hence on substituting x = 0 and y = 6 in its equation, we get 6 = a(-4)²-9 or, 16a= 6+9 =15 so that a = 15/16. Hence, the equation of the parabola is y =15/16(x-4)²-9. We know that, a function f(x) is symmetric about the origin if f(-x) = -f(x). It is not the case here. Hence the graph is not symmetric about the origin.