Evaluate the function [tex]f(x)=5 x+1[/tex] at the given values of the independent variable and simplify.
a. [tex]f(-1)[/tex]
b. [tex]f(x+5)[/tex]
c. [tex]f(-x)[/tex]
a. [tex]f(-1)=[/tex] (Simplify your answer.)
Answer (2)
The evaluations of the function f ( x ) = 5 x + 1 yield f ( − 1 ) = − 4 , f ( x + 5 ) = 5 x + 26 , and f ( − x ) = − 5 x + 1 . Each evaluation involves substituting the specified input into the function and simplifying. This process is important for understanding how functions work.
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Substitute x = − 1 into f ( x ) = 5 x + 1 to find f ( − 1 ) .
Calculate f ( − 1 ) = 5 ( − 1 ) + 1 = − 5 + 1 = − 4 .
Substitute x = x + 5 into f ( x ) = 5 x + 1 to find f ( x + 5 ) .
Calculate f ( x + 5 ) = 5 ( x + 5 ) + 1 = 5 x + 25 + 1 = 5 x + 26 .
Substitute x = − x into f ( x ) = 5 x + 1 to find f ( − x ) .
Calculate f ( − x ) = 5 ( − x ) + 1 = − 5 x + 1 .
The final answer is − 4 .
Explanation
Understanding the Problem We are given the function f ( x ) = 5 x + 1 and asked to evaluate it at three different values: x = − 1 , x = x + 5 , and x = − x . This involves substituting each of these values into the function and simplifying the resulting expression.
Evaluating f(-1) a. To find f ( − 1 ) , we substitute x = − 1 into the function: f ( − 1 ) = 5 ( − 1 ) + 1 = − 5 + 1 = − 4.
Evaluating f(x+5) b. To find f ( x + 5 ) , we substitute x = x + 5 into the function: f ( x + 5 ) = 5 ( x + 5 ) + 1 = 5 x + 25 + 1 = 5 x + 26.
Evaluating f(-x) c. To find f ( − x ) , we substitute x = − x into the function: f ( − x ) = 5 ( − x ) + 1 = − 5 x + 1.
Final Answer Therefore, we have: a. f ( − 1 ) = − 4 b. f ( x + 5 ) = 5 x + 26 c. f ( − x ) = − 5 x + 1 The answer to part a is -4.
Examples
Understanding function evaluation is crucial in many real-world applications. For instance, in physics, if f ( x ) represents the distance an object travels in time x , then evaluating f ( x ) at different times helps us understand the object's position at those times. Similarly, in economics, if f ( x ) represents the cost of producing x items, then evaluating f ( x ) at different production levels helps us understand the cost implications. Function evaluation is a fundamental tool for modeling and analyzing relationships between variables in various fields.
