Question

Find \(f(-1)\) if \(f(x)=2 x^{2}-x\).

Short answer

The value of \( f(-1) \) is \( 3 \).

Step-by-step solution

Identify the function and the input

The given function is \( f(x) = 2 x^2 - x \). We need to evaluate this function at \( x = -1 \).

Substitute the input into the function

Replace \( x \) with \( -1 \) in the function. Therefore, \( f(-1) = 2(-1)^2 - (-1) \).

Simplify the exponent

Calculate the exponent first. \( (-1)^2 = 1 \). So, \( f(-1) = 2 \times 1 - (-1) \).

Perform multiplication

Multiply \( 2 \) by \( 1 \). Therefore, \( f(-1) = 2 - (-1) \).

Simplify the expression

Simplify the subtraction of a negative number: \( 2 - (-1) = 2 + 1 = 3 \).

Conclusion

Thus, \( f(-1) = 3 \).

Key concepts

Substitution

To solve functions, substitution is a critical concept. In the given exercise, we need to find the value of the function \( f(x) \) when \( x = -1 \). This involves replacing every occurrence of the variable \( x \) in the function with \( -1 \). For instance, in \( f(x)=2x^2-x \), we substitute \( -1 \) for \( x \), resulting in \( f(-1) = 2(-1)^2 - (-1) \). By mastering substitution, you can solve numerous function-related problems quickly and effectively. Remember to follow the order of operations (PEMDAS) during substitution.

Exponents

Exponents play a vital role in algebra. In the given problem, we encountered an exponent situation while evaluating \( (-1)^2 \). Breaking this down, we understand that an exponent tells us how many times to multiply the base (which is \(-1\) in this case) by itself. Therefore, \( (-1)^2 = (-1) \times (-1) = 1 \). Calculating exponents first ensures you remain aligned with mathematical principles.

Simplification

Simplification makes expressions more manageable. Once we substituted \( x = -1 \) and calculated the exponent, we had \( f(-1) = 2 \times 1 - (-1) \). The next simplification step involves performing the operations. Multiplying \( 2 \times 1 \) gives us \( 2 \). Simplifying further, subtracting a negative number is the same as adding its positive counterpart. Thus, \( 2 - (-1) \) simplifies to \( 2 + 1 \). This results in \( 3 \). Simplification ensures clarity and accuracy in our solutions.

Basic Algebra

Basic algebra involves using basic operations and principles to manipulate and solve equations. When evaluating \( f(-1) \) for \( f(x) = 2x^2 - x \), we apply basic algebraic steps. This includes substituting \( -1 \) for \( x \), simplifying exponents, performing multiplications, and simplifying the resulting expression to get to the answer. Practicing basic algebra helps in solving more complex problems with confidence. Follow each step methodically to avoid mistakes.