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Chapter 2: Problem 4

If \(f(x)=2 x^{2}+4 x+3\), what is: (i) \(f(a)\) (ii) \(f\left(y^{2}\right)\)

Short Answer

Expert verified
(i) \( f(a) = 2a^2 + 4a + 3 \); (ii) \( f(y^2) = 2y^4 + 4y^2 + 3 \).

Step by step solution

01

Understanding the Function

The given function is \( f(x) = 2x^2 + 4x + 3 \). It is a quadratic function in terms of \( x \). The function calculates values based on the input \( x \). Our task is to find \( f(a) \) and \( f(y^2) \) by substituting \( x \) with \( a \) and \( y^2 \) respectively.
02

Substituting 'a' for 'x' in the Function

To find \( f(a) \), substitute \( a \) in place of \( x \) in the function. This gives us:\[ f(a) = 2a^2 + 4a + 3 \] There are no further simplifications since \( a \) is a variable.
03

Substituting 'y^2' for 'x' in the Function

To find \( f(y^2) \), substitute \( y^2 \) in place of \( x \) in the function. This gives us:\[ f(y^2) = 2(y^2)^2 + 4(y^2) + 3 \] Now, simplify \( 2(y^2)^2 \) to get:\[ 2y^4 \] The expression simplifies to:\[ 2y^4 + 4y^2 + 3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Substitution
Polynomial substitution is a crucial method in algebra that allows us to evaluate functions by replacing their variables with specific values or expressions. For a quadratic function like \( f(x) = 2x^2 + 4x + 3 \), substituting a variable involves replacing \( x \) with another value or expression. This helps us understand how the function behaves for different inputs. When you encounter a problem requiring you to find \( f(a) \), you substitute \( a \) in place of \( x \), resulting in \( f(a) = 2a^2 + 4a + 3 \). Similarly, for \( f(y^2) \), replace \( x \) with \( y^2 \):
  • Simplify \( (y^2)^2 \) to obtain \( y^4 \)
  • Continue with \( f(y^2) = 2y^4 + 4y^2 + 3 \)
These substitutions allow us to see how the function transforms for specific changes in its input.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and mathematical operations. In a function like \( f(x) = 2x^2 + 4x + 3 \), each part of the expression works together to define the function.Understanding expressions help in:
  • Identifying coefficients, such as 2 for \( x^2 \) and 4 for \( x \)
  • Recognizing constant terms, like 3
  • Simplifying and rearranging terms when needed
When we perform tasks like polynomial substitution, we rely on our understanding of these expressions and their components to accurately transform and simplify the functions.
Function Evaluation
Function evaluation is the process of calculating the value of a function for a specific input. In this exercise, the evaluation involves substituting variables into the function to find outputs. This helps in understanding the nature and behavior of the function.For example:
  • Evaluate \( f(a) \) by substituting \( x \) with \( a \) to get \( 2a^2 + 4a + 3 \)
  • Evaluate \( f(y^2) \) by substituting \( x \) with \( y^2 \), leading to \( 2y^4 + 4y^2 + 3 \)
Evaluating functions is not only about substitution but also about ensuring each step accurately represents the transformation or simplification required. It allows students to assess how changes in input variables influence the overall value of the function.

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Most popular questions from this chapter

Simplify if possible:\(\frac{x^{2}-9}{x^{2}+5 x+6}\)

Given that \(x^{2}-1\) is a factor of the quartic \(x^{4}-5 x^{3}+5 x^{2}+5 x-6\), (i) find the roots, (ii) sketch the graph.

Express in terms of partial fractions:\(\frac{x^{2}+2 x-1}{(x-1)^{2}(x+2)}\)

Solve the simultaneous equations:\(2 x-y=2, x^{2}-x y+y^{2}=1\)

Find \(x\) as a function of \(y=$$y=\frac{x}{1-x}\)
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