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 solve this problem, we need to substitute the functions g(x) and h(x) into f(x) and analyze the resulting composite functions.
a. f(g(x)) = f(x - 3)
To determine the transformation applied to function f by composing it with g, we replace x in f(x) with g(x), which is x - 3. This means that every x in f(x) will be replaced by (x - 3). So, the composite function f(g(x)) will be f(x - 3).
b. f(h(x)) = f(1/2x)
To determine the transformation applied to function f by composing it with h, we replace x in f(x) with h(x), which is 1/2x. This means that every x in f(x) will be replaced by 1/2x. So, the composite function f(h(x)) will be f(1/2x).
c. Plotting the graphs:
To plot the graphs of f, fog, and foh, we need to understand the individual functions first.
1. Function f(x) = x^2 - 2 (for -2 ≤ x ≤ 2):
The graph of f(x) is a parabola opening upwards with its vertex at (0, -2). The x-values are limited to the interval [-2, 2], and the y-values increase as x moves away from zero.
2. Function g(x) = x - 3 (for all real values of x):
The graph of g(x) is a straight line with slope 1 and a y-intercept at -3. It extends infinitely in both directions.
3. Function h(x) = (1/2)x (for all real values of x):
The graph of h(x) is a straight line with a slope of 1/2 and passes through the origin (0,0). It also extends infinitely in both directions.
Next, we'll plot the composite functions:
- fog = f(g(x)):
To determine fog, we substitute g(x) = x - 3 into f(x): f(g(x)) = f(x - 3). This means that we shift the graph of f(x) three units to the right. The new vertex will be at (3, -2). The parabolic shape remains the same.
- foh = f(h(x)):
To determine foh, we substitute h(x) = (1/2)x into f(x): f(h(x)) = f(1/2x). This means that we stretch or compress the graph of f(x) horizontally by a factor of 2. The vertex remains at (0, -2), but the intervals [-2, 2] become [-4, 4].
By considering the transformations applied to function f, the resulting graphs of fog and foh should confirm our conclusions. The shape and vertex positions of fog and foh will match our explanation.
Please note that without specific values for x, it is not possible to provide the exact coordinates of the graphs. The above explanation provides a general understanding of the transformations applied.