To determine which of the given functions is linear, we need to analyze each function separately. A linear function has the general form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is raised to the power of 1.
Let's examine each function:
### (a) [tex]\( x y = 12 \)[/tex]
This equation suggests a relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. To determine if it is linear, we can try to express it in the form of [tex]\( y = mx + b \)[/tex].
Rewrite the equation:
[tex]\[ xy = 12 \][/tex]
[tex]\[ y = \frac{12}{x} \][/tex]
Since [tex]\( y \)[/tex] is equal to [tex]\(\frac{12}{x}\)[/tex], we observe that [tex]\( y \)[/tex] is not a linear function of [tex]\( x \)[/tex] because [tex]\( x \)[/tex] is in the denominator. For a function to be linear, [tex]\( y \)[/tex] must be a polynomial of degree 1 in [tex]\( x \)[/tex]. Hence, this is not a linear function.
### (b) [tex]\( y = x^5 - x^4 \)[/tex]
Here, [tex]\( y \)[/tex] is expressed directly as a function of [tex]\( x \)[/tex].
Examine the powers of [tex]\( x \)[/tex] in the equation:
[tex]\[ y = x^5 - x^4 \][/tex]
In this case, [tex]\( y \)[/tex] involves terms like [tex]\( x^5 \)[/tex] and [tex]\( x^4 \)[/tex], both of which are powers of [tex]\( x \)[/tex] greater than 1. For a function to be linear, the highest power of [tex]\( x \)[/tex] must be 1. Thus, this function is not linear.
### (c) [tex]\( 3x + 5y = 9 \)[/tex]
This is a linear equation in its standard form. To see it more clearly, we can isolate [tex]\( y \)[/tex] on one side of the equation:
[tex]\[ 3x + 5y = 9 \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 5y = -3x + 9 \][/tex]
Divide every term by 5:
[tex]\[ y = -\frac{3}{5}x + \frac{9}{5} \][/tex]
This equation is now in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m = -\frac{3}{5} \)[/tex] and [tex]\( b = \frac{9}{5} \)[/tex]. Since this is the form of a linear function, this function is indeed linear.
### (d) [tex]\( y - 2 = x(x + 1) \)[/tex]
First, simplify the right side of the equation:
[tex]\[ y - 2 = x(x + 1) \][/tex]
[tex]\[ y - 2 = x^2 + x \][/tex]
Add 2 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = x^2 + x + 2 \][/tex]
This results in a quadratic term [tex]\( x^2 \)[/tex], which indicates the presence of a polynomial of degree 2. For the function to be linear, the highest power of [tex]\( x \)[/tex] must be 1. Therefore, this function is not linear.
### Conclusion
Among the given functions, only function (c) [tex]\( 3x + 5y = 9 \)[/tex] is linear. The other functions either have terms with powers of [tex]\( x \)[/tex] greater than 1 or involve [tex]\( x \)[/tex] in the denominator, making them non-linear.
So, the linear function is:
[tex]\[ \boxed{3} \][/tex]
