Final answer:
The statement about the function t(x) = -6(2)^x is that it is an exponential function with exponential growth reflected across the x-axis, with no x-intercepts and a y-intercept of -6.
Step-by-step explanation:
The function in question, t(x) = -6(2)x, is an exponential function where the base is 2 and the exponential expression is multiplied by -6. Since the base, 2, is positive and greater than 1, the function represents exponential growth. However, the negative sign in front of the 6 indicates that the function will be reflected across the x-axis, and therefore, the graph of t(x) will decrease as x increases; every successive y-value will be a negative value that is twice the magnitude of the previous y-value when x is incremented by 1.
Furthermore, the function does not have any x-intercepts because an exponential function (other than when the base is 1) never equals zero, and thus the graph will never cross the x-axis. The y-intercept of the function can be found by evaluating t(x) at x=0, which yields t(0) = -6(2)0 = -6.
To determine which statement about the function is true, we need to evaluate each statement using the given function.
Let's go through the statements one by one:
- t(-1) = -6(2)^(-1) = -6(1/2) = -3. Therefore, the first statement is false.
- t(0) = -6(2)^0 = -6(1) = -6. Therefore, the second statement is false.
- t(1) = -6(2)^1 = -6(2) = -12. Therefore, the third statement is false.
- t(2) = -6(2)^2 = -6(4) = -24. Therefore, the fourth statement is false.
- t(3) = -6(2)^3 = -6(8) = -48. Therefore, the fifth statement is false.
None of the statements are true.
Therefore, the correct answer is that no statement about t(x) = -6(2)^x is true.