Determine the equation for g(x)and leave the answer in the form y = ... Determine the equation of g−1(x)…

Question

Determine the equation for g(x)and leave the answer in the form y = ... Determine the equation of g−1(x). Give the answer in the form x =... Write down an equation for the asymptote of g−1(x).

ScanSolve AI Tutor · Accepted Answer

1.1 Given the function: f(x) = −x² − 2x² + 5x + 6 Step 1: Simplify the function. f(x) = −3x² + 5x + 6; Step 2: To find the roots, set f(x) = 0. −3x² + 5x + 6 = 0 Step 3: Multiply by −1 to make the leading coefficient positive. 3x² − 5x − 6 = 0; Step 4: Use the quadratic formula x = (−b ± √(b² − 4ac))/(2a) with a=3, b=−5, c=−6. x = (−(−5) ± √((−5)² − 4(3)(−6)))/(2(3)); x = (5 ± √(25 + 72))/(6); x = (5 ± √(97))/(6) The final answers are: x = (5 + √(97))/(6) or x = (5 − √(97))/(6) 1.2 Given the function f(x) = x³ + ax² + bx + 8 is divisible by x² − x − 2. Step 1: Factorize the divisor x² − x − 2. x² − x − 2 = (x−2)(x+1); Step 2: Since f(x) is divisible by (x−2)(x+1), it means x=2 and x=−1 are roots of f(x). Therefore, f(2) = 0 and f(−1) = 0. Step 3: Substitute x=2 into f(x) and set it to 0. f(2) = (2)³ + a(2)² + b(2) + 8 = 0; 8 + 4a + 2b + 8 = 0; 4a + 2b + 16 = 0 Divide by 2: 2a + b + 8 = 0 ⇒ 2a + b = −8 (Equation 1); Step 4: Substitute x=−1 into f(x) and set it to 0. f(−1) = (−1)³ + a(−1)² + b(−1) + 8 = 0; −1 + a − b + 8 = 0; a − b + 7 = 0 ⇒ a − b = −7 (Equation 2) Step 5: Solve the system of linear equations (Equation 1 and Equation 2). Add Equation 1 and Equation 2: (2a + b) + (a − b) = −8 + (−7); 3a = −15; a = −5; Step 6: Substitute a=−5 into Equation 2 to find b. −5 − b = −7; −b = −7 + 5; −b = −2; b = 2 The final answers are: a = −5, b = 2 1.3 Given the function f(x) = x³ − kx² − x + 13. When divided by x−3, the remainder is 10. Step 1: Apply the Remainder Theorem, which states that if a polynomial f(x) is divided by x−c, the remainder is f(c). In this case, c=3 and the remainder is 10, so f(3) = 10. Step 2: Substitute x=3 into f(x) and set it equal to 10. f(3) = (3)³ − k(3)² − (3) + 13 = 10; 27 − 9k − 3 + 13 = 10 Step 3: Simplify and solve for k. 37 − 9k = 10; −9k = 10 − 37; −9k = −27; k = (−27)/(−9); k = 3 The final answer is: k = 3

Determine the equation for g(x)and leave the answer in the form y = ... Determine the equation of g-1(x). Give the answer in the form x =... Write down an equation for the asymptote of g-1(x).

Answers

Step-by-step solution

Part 1.1· 4 steps

Simplify the function.

To find the roots, set $f(x) = 0$ .

Multiply by $-1$ to make the leading coefficient positive.

Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=3$ , $b=-5$ , $c=-6$ .

Part 1.2· 6 steps
$a = -5, b = 2$

Factorize the divisor $x^2 - x - 2$ .

Since $f(x)$ is divisible by $(x-2)(x+1)$ , it means $x=2$ and $x=-1$ are roots of $f(x)$ .

Substitute $x=2$ into $f(x)$ and set it to 0.

Substitute $x=-1$ into $f(x)$ and set it to 0.

Solve the system of linear equations (Equation 1 and Equation 2).

Substitute $a=-5$ into Equation 2 to find $b$ .

Part 1.3· 3 steps
$k = 3$

Apply the Remainder Theorem, which states that if a polynomial $f(x)$ is divided by $x-c$ , the remainder is $f(c)$ .

Substitute $x=3$ into $f(x)$ and set it equal to 10.

Simplify and solve for $k$ .

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More Mathematics Questions

Determine the equation for g(x)and leave the answer in the form y = ... Determine the equation of g-1(x). Give the answer in the form x =... Write down an equation for the asymptote of g-1(x).

Answers

Step-by-step solution

Part 1.1· 4 steps

Simplify the function.

To find the roots, set f(x)=0f(x) = 0f(x)=0.

Multiply by −1-1−1 to make the leading coefficient positive.

Use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​ with a=3a=3a=3, b=−5b=-5b=−5, c=−6c=-6c=−6.

Part 1.2· 6 stepsa=−5,b=2a = -5, b = 2a=−5,b=2

Factorize the divisor x2−x−2x^2 - x - 2x2−x−2.

Since f(x)f(x)f(x) is divisible by (x−2)(x+1)(x-2)(x+1)(x−2)(x+1), it means x=2x=2x=2 and x=−1x=-1x=−1 are roots of f(x)f(x)f(x).

Substitute x=2x=2x=2 into f(x)f(x)f(x) and set it to 0.

Substitute x=−1x=-1x=−1 into f(x)f(x)f(x) and set it to 0.