Let f(x) = e^x – 2 + 3
a. Describe in words how the graph of f can be obtained from the graph of y = e^x.
b. What is the domain of f?
c. What is the range of f?
d. What is the y-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth).
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(a) Describe in words how the graph of f can be obtained from the graph of y = e^x.
The graph of f can be obtained from the graph of y = e^x by off setting the graph by one.
(b) What is the domain of f? x= {x|-&<x<+&, xEReal} X is the element of real numbers this means Infinite.
Domain of f is all real values of f.
(c) What is the range of f? The range is all the real numbers except three.
(d) What is the y-intercept? State the approximation to 2 decimal places (i.e., the nearest hundredth).
y intercept is when x is zero. e^-2 + 3 or about 1/e^2 + 3 =approximately 1/6 + 3 + 3.15 checking that in a calculator:???????????????????
IF X=0 THEN Y=e^ (-2) +3
e=2.7
2.7^-2+3=1/2.7^2+3= 3.14
a. To obtain the graph of f(x) from the graph of y = e^x, you need to make two transformations. First, shift the graph of y = e^x vertically downwards by 2 units. This can be done by subtracting 2 from every y-coordinate of the original graph. Secondly, shift the graph obtained in the previous step vertically upwards by 3 units. This can be done by adding 3 to every y-coordinate of the modified graph. This transformation results in the graph of f(x) = e^x - 2 + 3.
b. The domain of f(x) is the set of all possible x-values for which the function is defined. In this case, since e^x is defined for all real numbers, there are no restrictions on the values x can take. Hence, the domain of f is the set of all real numbers, or (-∞, +∞).
c. The range of f(x) is the set of all possible y-values that the function can take. In this case, since the exponential function e^x is always positive, the lowest value f(x) can take is when e^x - 2 is equal to 0, which occurs when x = ln(2) (approximately 0.69). Adding 3 to this minimum value gives the lowest possible value of f(x), which is approximately 3 - 2 + 3 = 4. Therefore, the range of f is [4, +∞).
d. The y-intercept represents the value of f(x) when x = 0, which can be found by substituting x = 0 into the function f(x). Thus, evaluating f(0) = e^0 - 2 + 3 = 1 - 2 + 3 = 2. Therefore, the y-intercept is approximately 2.00.