Chapter 33: Problem 2
Given that \(f(x)=5 x^{2}+x-7\) determine: (i) \(f(2) \div f(1)\) (ii) \(f(3+a)\) (iii) \(f(3+a)-f(3)\) (iv) \(\frac{f(3+a)-f(3)}{a}\)
Short Answer
Expert verified
(i) -15, (ii) 5a^2 + 31a + 41, (iii) 5a^2 + 31a, (iv) 5a + 31
Step by step solution
01
Evaluate f(2)
Substitute \( x = 2 \) into the function \( f(x) = 5x^2 + x - 7 \). Calculate: \[f(2) = 5(2)^2 + (2) - 7 = 20 + 2 - 7 = 15\]
02
Evaluate f(1)
Substitute \( x = 1 \) into the function \( f(x) = 5x^2 + x - 7 \). Calculate: \[f(1) = 5(1)^2 + (1) - 7 = 5 + 1 - 7 = -1\]
03
Calculate f(2) รท f(1)
Use the values from Step 1 and Step 2. Calculate: \[f(2) \, \div \, f(1) = \frac{15}{-1} = -15\]
04
Substitute 3+a into the Function
Replace \( x \) in \( f(x) = 5x^2 + x - 7 \) with \( 3+a \) and simplify:\[f(3+a) = 5(3+a)^2 + (3+a) - 7\]Expand \( (3+a)^2 \):\[= 5(9 + 6a + a^2) + 3 + a - 7\]Distribute and simplify:\[= 45 + 30a + 5a^2 + 3 + a - 7 = 5a^2 + 31a + 41\]
05
Calculate f(3)
Substitute \( x = 3 \) into the function \( f(x) = 5x^2 + x - 7 \) and simplify:\[f(3) = 5(3)^2 + 3 - 7 = 45 + 3 - 7 = 41\]
06
Calculate f(3+a) - f(3)
Use the expressions from Step 4 and Step 5. Subtract the two:\[f(3+a) - f(3) = (5a^2 + 31a + 41) - 41 = 5a^2 + 31a\]
07
Simplify \\(\\frac{f(3+a) - f(3)}{a}\\)
Divide the expression obtained in Step 6 by \( a \):\[\frac{f(3+a) - f(3)}{a} = \frac{5a^2 + 31a}{a} = 5a + 31\] Simplify by canceling the \( a \) from numerator and denominator.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Functions
Polynomial functions are a type of function that includes terms built from variables raised to various powers, added or subtracted together. They look like this:
- Each term has a coefficient and a variable raised to a non-negative integer power.
- The function in this exercise is a quadratic polynomial, meaning its highest power is a square, as in \(5x^2 + x - 7\).
- The first term is \(5x^2\), which is called a quadratic term.
- The second term is \(x\), the linear term, and lastly,
- \(-7\), the constant term.
Substitution Method
The substitution method is a technique used to evaluate polynomial functions. Essentially, you plug a number into the function wherever you see the variable. Let's walk through the steps of this process in the given exercise:
- Step 1 was to find \(f(2)\): Substitute \(x = 2\) into the function. This substitution changes each instance of \(x\) in the equation to 2, leading to a straightforward calculation to evaluate the function value.
- Similarly, for \(f(1)\), we put \(x = 1\) into the function \(5x^2 + x - 7\).
- To find \(f(3+a)\), replace \(x\) with \(3+a\), which involves a bit more algebra as it requires expanding and simplifying.
Simplification
Simplification is the process of reducing a mathematical expression to its most concise form without changing its value. This operation requires understanding and applying algebraic rules and properties. Hereโs how it was used in this exercise:
- For the expression \(f(3+a) = 5(3+a)^2 + (3+a) - 7\), expand the square and combine all like terms. This reduces the expression to a clearer form: \(5a^2 + 31a + 41\).
- We then used this to calculate \(f(3+a) - f(3)\), needing further simplification to \(5a^2 + 31a\).
- Finally, simplifying \(\frac{f(3+a) - f(3)}{a}\) involved canceling out the common factor of \(a\) in the numerator and denominator, leaving us with \(5a + 31\).