Horizontal Translation and dialation Problem : let f, g, and h be defined f(x)= x2 -2< or equal to x < or equal to 2

g(x)=x-3 for all real valuse of x
h(x) = 1/2 x for all real values of x
a. f(g(x))= f(x-3). What transformatioon is applied to function f by composing it with g?
b. f(h(x))= f(1/2x). What tranformation is applied to function f by composing it with h?
c. Plot the graphs f, fog and foh. Sketch the results. Do the graphs confirm your conclusions in part a and b

To answer these questions, let's first understand how to compose functions, and then we can analyze the transformations applied to function f.

Function Composition:
When we compose two functions, say f(x) and g(x), written as (f ∘ g)(x), it means that we first apply the function g to the input x and then take the result of g(x) and plug it into function f.

a. f(g(x)) = f(x - 3)
In this case, we are composing functions f and g. It means that we first apply the function g(x) = x - 3, and then take the result of g(x) and plug it into function f(x) = x^2 - 2.

To analyze the transformation applied to function f, we need to understand how g(x) affects the input to f(x). In this case, g(x) subtracts 3 from x. So, the graph of f(g(x)) is obtained by shifting the graph of f(x) three units to the right.

b. f(h(x)) = f(1/2x)
Similarly, in this case, we are composing functions f and h. The function h(x) = (1/2)x multiplies the input x by 1/2. So, the graph of f(h(x)) is obtained by horizontally stretching the graph of f(x) by a factor of 2.

Now, let's plot the graphs of f(x), f(g(x)), and f(h(x)) to visually confirm our conclusions:

- Graph of f(x):
The given function f(x) = x^2 - 2 is a parabola centered at (0, -2) with the vertex pointing upward.

- Graph of f(g(x)):
To obtain the graph of f(g(x)), we need to shift the graph of f(x) three units to the right. This means that every point on the graph of f(x) will move to the right by three units. So, the vertex of f(x), which is at (0, -2), will move to (3, -2).

- Graph of f(h(x)):
To obtain the graph of f(h(x)), we need to horizontally stretch the graph of f(x) by a factor of 2. This means that every point on the graph of f(x) will be spread out horizontally. Since the function f(x) is symmetric with respect to the y-axis, the stretched graph will still be symmetric with respect to the y-axis.

Plotting the three graphs on the same coordinate system, we will see the confirmation of the conclusions made in parts a and b.

I hope this explanation helps!