If g(x)>0 for all real values of x, which of the following could be the function g?
a) g(x) = x + 1
b) g(x) = x - 1
c) g(x) = x^2 + 1
d) g(x) = x^2 - 1
e) g(x)= x^3 1
x^2 is the only one that is always positive but x^2-1 could be negative so you only have one choice left.
Well, it sounds like g(x) is always positive. That's pretty optimistic of g(x), always looking on the bright side! So, let's see which of these options keeps that optimistic spirit going.
Let's go through them one by one:
a) g(x) = x + 1
Hmm, if we plug in a negative value for x, like -2, we get g(-2) = -2 + 1 = -1. So, unfortunately, this one doesn't make the cut. Negative values aren't really the most positive of things, are they?
b) g(x) = x - 1
Let's try the same negative value for x: g(-2) = -2 - 1 = -3. Oops, same problem as before. Negative values are raining on this parade.
c) g(x) = x^2 + 1
Ah, now we're getting somewhere. If we look at the square of any real number, it's always positive or zero. And adding 1 to a positive number keeps it positive. Positive squared positivity? Sign me up!
d) g(x) = x^2 - 1
Oh dear, this one's a bit gloomy. If we plug in x = 0, we get g(0) = 0 - 1 = -1. Negative values are always raining on this clown parade!
e) g(x) = x^3 - 1
Hmmm, let's see. If we pick x = 1, we get g(1) = 1^3 - 1 = 0. And zero isn't really a positive number, is it? This option might need a little more time in the positivity training camp.
So, after all that, the function g(x) = x^2 + 1 is the only one that consistently looks at life through its pink-colored clown wig and stays positive for all real values of x.
To determine which functions could be g(x) such that g(x) is greater than 0 for all real values of x, we can analyze each function option one by one.
a) g(x) = x + 1:
Let's test some values for x to see if g(x) is positive for all real values:
- When x = 0, g(x) = 0 + 1 = 1, which is greater than 0.
- When x = -2, g(x) = -2 + 1 = -1, which is less than 0.
Therefore, g(x) = x + 1 does not satisfy the condition.
b) g(x) = x - 1:
Testing some values:
- When x = 0, g(x) = 0 - 1 = -1, which is less than 0.
- When x = 2, g(x) = 2 - 1 = 1, which is greater than 0.
Thus, g(x) = x - 1 also does not satisfy the condition.
c) g(x) = x^2 + 1:
Testing some values:
- When x = 0, g(x) = 0^2 + 1 = 1, which is greater than 0.
- When x = 2, g(x) = 2^2 + 1 = 5, which is also greater than 0.
Therefore, g(x) = x^2 + 1 satisfies the condition.
d) g(x) = x^2 - 1:
Testing some values:
- When x = 0, g(x) = 0^2 - 1 = -1, which is less than 0.
- When x = 2, g(x) = 2^2 - 1 = 3, which is greater than 0.
Thus, g(x) = x^2 - 1 does not satisfy the condition.
e) g(x) = x^3 - 1:
Testing some values:
- When x = 0, g(x) = 0^3 - 1 = -1, which is less than 0.
- When x = 2, g(x) = 2^3 - 1 = 7, which is greater than 0.
Therefore, g(x) = x^3 - 1 does not satisfy the condition.
Based on our analysis, the only function that satisfies the condition g(x) > 0 for all real values of x is g(x) = x^2 + 1. Therefore, the correct answer is c) g(x) = x^2 + 1.
To determine which of the given functions could satisfy the condition g(x) > 0 for all real values of x, we need to analyze each function.
a) g(x) = x + 1
For this function, since the coefficient of x is positive, the function will always be increasing for any value of x. This means that g(x) > 0 for all real values of x.
b) g(x) = x - 1
Similar to the previous function, the coefficient of x is positive, so the function will also always be increasing. Therefore, g(x) > 0 for all real values of x.
c) g(x) = x^2 + 1
This is a quadratic function, and since the coefficient of the x^2 term is positive, the graph of this function will open upward. However, since the lowest point on the graph is at (0, 1), there are values of x for which g(x) = 0 or g(x) < 0. Therefore, g(x) > 0 does not hold true for all real values of x.
d) g(x) = x^2 - 1
Similar to the previous function, this quadratic function also opens upward. However, in this case, the lowest point on the graph is at (0, -1). So again, there are values of x for which g(x) = 0 or g(x) < 0, and therefore, g(x) > 0 does not hold true for all real values of x.
e) g(x) = x^3 - 1
For this function, since the coefficient of the x^3 term is positive, the graph will always be increasing for any value of x. This means that g(x) > 0 for all real values of x.
Therefore, the functions that could satisfy the condition g(x) > 0 for all real values of x are option a) g(x) = x + 1 and option b) g(x) = x - 1.