Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine 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) 梅 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.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
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. Here鈥檚 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 \(5a + 31\).
