Select all the correct statements. (For this problem, we assume that an exponential function is of the form ka^x where a > 0)
(A) The following function is exponential: y = 2^x
(B) The following function is exponential: y = x^2
(C) The following function is exponential: y = sqrt(x)
(D) The following function is exponential: y = cube root of x
(E) The following function is exponential: y = 1/x
(F) The following function is exponential: y = 8x - 4
Also, solve these exponential equations:
(A) 3^(2y) = 3^5
(B) 9^(3u + 5) = 3^8
(C) 8^(4x - 1) = 64^(5x)
Also, prove the following statements:
(A) If the graph of an exponential function is reflected in the x-axis, then we obtain the graph of another exponential function
(B) If the graph of an exponential function is reflected in the y-axis, then we obtain the graph of another exponential function
(C) If the graph of an exponential function is translated vertically, then we obtain the graph of another exponential function.
(D) If the graph of an exponential function is translated horizontally, then we obtain the graph of another exponential function
To identify whether a function is exponential, we need to check if it follows the form of ka^x, where a is greater than 0.
(A) The function y = 2^x is exponential because it follows the form ka^x, where k = 1 and a = 2. So this statement is correct.
(B) The function y = x^2 is not exponential because it does not follow the form ka^x. It is a quadratic function. So this statement is incorrect.
(C) The function y = sqrt(x) is not exponential because it does not follow the form ka^x. It is a square root function. So this statement is incorrect.
(D) The function y = cube root of x is not exponential because it does not follow the form ka^x. It is a cube root function. So this statement is incorrect.
(E) The function y = 1/x is not exponential because it does not follow the form ka^x. It is a reciprocal function. So this statement is incorrect.
(F) The function y = 8x - 4 is not exponential because it does not follow the form ka^x. It is a linear function. So this statement is incorrect.
Now let's solve the exponential equations:
(A) 3^(2y) = 3^5
To solve this equation, we can equate the exponents:
2y = 5
Dividing both sides by 2:
y = 5/2
So the solution is y = 2.5.
(B) 9^(3u + 5) = 3^8
We can rewrite 9 as 3^2:
(3^2)^(3u + 5) = 3^8
Using the power of a power rule, we can multiply the exponents:
3^(6u + 10) = 3^8
Since the bases are the same, we can equate the exponents:
6u + 10 = 8
Subtracting 10 from both sides:
6u = -2
Dividing both sides by 6:
u = -2/6
Simplifying:
u = -1/3
So the solution is u = -1/3.
(C) 8^(4x - 1) = 64^(5x)
Since 64 is 8^2, we can rewrite the equation as:
8^(4x - 1) = (8^2)^(5x)
Using the power of a power rule, we can multiply the exponents:
8^(4x - 1) = 8^(2 * 5x)
Since the bases are the same, we can equate the exponents:
4x - 1 = 10x
Subtracting 4x from both sides:
-1 = 6x
Dividing both sides by 6:
x = -1/6
So the solution is x = -1/6.
Now let's prove the given statements:
(A) If the graph of an exponential function is reflected in the x-axis, then we obtain the graph of another exponential function.
To prove this, we can consider the general form of an exponential function, y = ka^x. When the graph is reflected in the x-axis, the y-values will change sign (positive becomes negative, and negative becomes positive), but the x-values remain the same. Therefore, the form of the function is still ka^x, so we have another exponential function.
(B) If the graph of an exponential function is reflected in the y-axis, then we obtain the graph of another exponential function.
To prove this, we can again consider the general form of an exponential function, y = ka^x. When the graph is reflected in the y-axis, the x-values change sign (positive becomes negative, and negative becomes positive), but the y-values remain the same. Therefore, the form of the function is still ka^x, so we have another exponential function.
(C) If the graph of an exponential function is translated vertically, then we obtain the graph of another exponential function.
To prove this, we can consider the general form of an exponential function, y = ka^x. When the graph is translated vertically, the vertical position of each point changes, but the shape and growth/decay characteristics remain the same. Therefore, the form of the function is still ka^x, so we have another exponential function.
(D) If the graph of an exponential function is translated horizontally, then we obtain the graph of another exponential function.
To prove this, we can consider the general form of an exponential function, y = ka^x. When the graph is translated horizontally, the horizontal position of each point changes, but the shape and growth/decay characteristics remain the same. Therefore, the form of the function is still ka^x, so we have another exponential function.
In summary, the correct statements are:
(A) The following function is exponential: y = 2^x
(A) 3^(2y) = 3^5 has the solution y = 2.5
(B) 9^(3u + 5) = 3^8 has the solution u = -1/3
(C) 8^(4x - 1) = 64^(5x) has the solution x = -1/6
(A) If the graph of an exponential function is reflected in the x-axis, then we obtain the graph of another exponential function
(B) If the graph of an exponential function is reflected in the y-axis, then we obtain the graph of another exponential function
(C) If the graph of an exponential function is translated vertically, then we obtain the graph of another exponential function
(D) If the graph of an exponential function is translated horizontally, then we obtain the graph of another exponential function