options 'd' and 'f
First, let's understand what a linear function is. A linear function is any function that can be written in the form y = mx + c, where m is the slope (expressing the 'steepness' of the line) and c is the y-intercept (where the line crosses the y-axis).
Let's examine each option individually to determine if it can be written in this form:
a. For y=3 √(x), the variable x is under a square root, which means this is not a linear function.
b. Looking at y=7^(x), the variable x is in the exponent. This also does not meet the criteria for a linear function.
c. In y=(2)/(x), the variable x is in the denominator of the fraction, so this function is not linear either.
d. For y=3.3, it can be written in the form y=mx+c where m=0 (the slope is 0) since there's no x variable, and c=3.3 (the y-intercept is 3.3). Thus, this is a linear function.
e. The equation (1)/(3)x^(2)-y=0 involves the variable x being squared, which indicates that it is not a linear function.
f. In y=-2(x+1), the function can be rewritten in the form y=mx+c where m=-2 (the slope is -2) and c=2 (the y-intercept is 2). Thus, it is linear.
To summarize, the only linear functions from the given options are y=3.3 and y=-2(x+1), which are options 'd' and 'f'.