2. The function f is defined as follows
f(x)={4+2x if x<0
{x^2 if x>0
a) Find the domain of the function
b) Locate any intercepts
c) Graph function
d) Based on graph find range
e) Is f continuous on its domain?
a) The domain of the function f is
To find the domain of the function f, we need to determine the set of all values of x for which the function is defined.
a) Domain:
In this case, the function f is defined for all real numbers except 0 because there is no rule defined for x=0. Therefore, the domain of the function f is (-∞, 0) U (0, ∞).
b) Intercepts:
To find the intercepts of the function, we need to determine the values of x for which f(x) equals zero.
For f(x) = 0,
If x < 0, then 4 + 2x = 0
Solving this equation, we get x = -2 as the intercept.
If x > 0, then x^2 = 0
Solving this equation, we get x = 0 as the intercept.
Therefore, the function has intercepts at x = -2 and x = 0.
c) Graph:
To graph the function f, we can plot points for different values of x, considering the defined rules for x < 0 and x > 0.
For x < 0, f(x) = 4 + 2x. We can choose some x values less than 0, substitute them into the equation, and plot the corresponding points on a graph.
For x > 0, f(x) = x^2. We can choose some x values greater than 0, substitute them into the equation, and plot the corresponding points on a graph.
Joining the plotted points, we can draw a graph of the function f.
d) Range:
To determine the range of the function, we need to analyze the graph.
From the graph, we can see that the range of the function f is [0, ∞) because the function takes all values greater than or equal to 0.
e) Continuity:
The function f is continuous on its domain except at x = 0, as there is a jump in the graph from 0 to a positive value.
In summary:
a) The domain of the function f is (-∞, 0) U (0, ∞).
b) The intercepts of the function are at x = -2 and x = 0.
c) The graph of the function f can be plotted by considering the rules for x < 0 and x > 0.
d) The range of the function is [0, ∞).
e) The function f is not continuous at x = 0.