Which of the following is a decreasing function?
a. y = 7^x - 2
b. y = 7^-x + 2
c. y = 3(5^x) - 5
d. y = 5(3^x) + 3
What function is the reflection of y = 10^x with respect to the y-axis?
a. y = -10^x
b. y = (-1/10)^x
c. y = 10^-x
d. y = -10^x
1b.
To determine whether a function is decreasing or not, we need to analyze the behavior of the function as the input (x-value) increases.
For option a. y = 7^x - 2, the base 7 raised to the power of any positive number x will always yield a positive result. And since subtracting a constant value (2) from a positive number does not change its sign, the function remains positive. Therefore, option a is not a decreasing function.
For option b. y = 7^-x + 2, the base 7 raised to a negative power results in a fraction or decimal value less than 1. As x increases, the function approaches 0, because the negative exponent causes the value to approach the reciprocal of a large number. Adding a constant value (2) to this small number remains positive, so option b is not a decreasing function.
For option c. y = 3(5^x) - 5, the base 5 raised to any positive number x will always yield a positive result. Multiplying a positive number by a positive constant value (3) yields a positive result. Subtracting a constant value (5) from a positive number does not change its sign. Therefore, option c is not a decreasing function.
For option d. y = 5(3^x) + 3, the base 3 raised to any positive number x will always yield a positive result. Multiplying a positive number by a positive constant value (5) yields a positive result. Adding a constant value (3) to a positive number does not change its sign. Therefore, option d is not a decreasing function.
None of the given options are decreasing functions.
Moving on to the second question, to find the reflection of a function with respect to the y-axis, we replace x with -x in the original function.
The original function is y = 10^x.
Replacing x with -x, we get y = 10^(-x).
So, the reflection of y = 10^x with respect to the y-axis is option c. y = 10^-x.