Pregunta
Determine whether the given function is linear. If the function is linear, express the function in the form \( f(x)=a x+b \). (If the functi \[ f(x)=\frac{4 x-5}{x} \] \( f(x)= \)
Ask by Curry Pierce. in the United States
Feb 07,2025
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The function \( f(x) = \frac{4x - 5}{x} \) simplifies to \( f(x) = 4 - \frac{5}{x} \). Since it includes a term with \( \frac{1}{x} \), it is not a linear function.
Solución
To determine if f(x) = (4x - 5)/x is linear, first simplify the expression:
f(x) = (4x)/x - 5/x = 4 - 5/x
A linear function has the form f(x) = ax + b, where a and b are constants. In the expression 4 - 5/x, the term -5/x is not of the form ax (since it involves the reciprocal of x, not x raised to the first power). Therefore, the function is not linear.
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To determine if the function \( f(x) = \frac{4x - 5}{x} \) is linear, we can simplify it. By dividing both terms in the numerator by \( x \), we get: \[ f(x) = \frac{4x}{x} - \frac{5}{x} = 4 - \frac{5}{x} \] Since the term \( \frac{5}{x} \) introduces a variable in the denominator, the function is not linear. A linear function should take the form \( f(x) = ax + b \) where \( a \) and \( b \) are constants. In this case, \( f(x) \) includes a term with \( x \) in the denominator, making it a non-linear function. Therefore, \( f(x) \) cannot be expressed as \( f(x) = ax + b \).
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