Totally confused! Got 5 more of these to do and need some guidence please.
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).
e) What is the horizontal asymptote?
I am not certain what your question is. Do you have any instructions?
Yes here they are
(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).
(e) What is the horizontal asymptote?
(a) To obtain the graph of f(x) = e^x–2 + 3 from the graph of y = e^x, you need to shift the graph of y = e^x two units to the left and three units up. In other words, you take each point (x, y) on the graph of y = e^x and move it two units left and three units up to get the corresponding point (x, y+3) on the graph of f(x) = e^x–2 + 3.
(b) The domain of a function represents all the possible x-values for which the function is defined. In this case, since the base function y = e^x is defined for all real numbers, there are no restrictions on the x-values for f(x) = e^x–2 + 3. Therefore, the domain of f is the set of all real numbers, or (-∞, ∞).
(c) The range of a function represents all the possible y-values it can take. In this case, when you add a constant term to the function y = e^x, it shifts the graph vertically but does not change the shape of the curve. Since e^x is always positive, adding a positive constant (3 in this case) to it will only shift the entire graph upwards. Therefore, the range of f is the set of all real numbers greater than or equal to 3, or [3, ∞).
(d) The y-intercept is the point where the graph of a function intersects the y-axis. To find the y-intercept, you set x to 0 in the equation f(x) = e^x–2 + 3:
f(0) = e^0–2 + 3 = e^0 + 1 = 1 + 1 = 2
Therefore, the y-intercept is approximately 2 (to the nearest hundredth).
(e) A horizontal asymptote describes the behavior of the graph as x approaches positive or negative infinity. In this case, since the base function y = e^x grows without bound as x approaches positive infinity and never approaches a fixed value, there is no horizontal asymptote for the function f(x) = e^x–2 + 3.