Solve the following questions: a) If h(x) = bx - 5 and h(4) = −1, find b. b) Simplify g(x) = (x2 - 1)/(x…

Question

Solve the following questions: a) If h(x) = bx - 5 and h(4) = −1, find b. b) Simplify g(x) = (x2 - 1)/(x − 1). c) State the domain of definition of g(x).

ScanSolve AI Tutor · Accepted Answer

Here's the solution for question 6: a) Step 1: Substitute x=4 into the function h(x) = bx − 5. h(4) = b(4) − 5; Step 2: Use the given information that h(4) = −1 to set up an equation. 4b − 5 = −1 Step 3: Add 5 to both sides of the equation. 4b = −1 + 5; 4b = 4 Step 4: Divide by 4 to solve for b. b = 4/4; b = 1 The value of b is 1. b) Step 1: Factor the numerator x² − 1 using the difference of squares formula, a² − b² = (a−b)(a+b). x² − 1 = (x−1)(x+1) Step 2: Substitute the factored numerator back into the expression for g(x). g(x) = ((x−1)(x+1))/(x−1) Step 3: Cancel out the common term (x−1) from the numerator and the denominator. g(x) = x+1 The simplified expression for g(x) is x+1. c) Step 1: Identify any values of x that would make the denominator of the original function g(x) = (x² − 1)/(x − 1) equal to zero. The denominator is x−1. x−1 = 0; x = 1 Step 2: The domain of definition includes all real numbers except for the values that make the denominator zero. Therefore, x ≠ 1. The domain of definition of g(x) is all real numbers except x=1.

Solve the following questions: a) If h(x) = bx − 5 and h(4) = -1, find b. b) Simplify g(x) = (x2 − 1)/(x - 1). c) State the domain of definition of g(x).

Answers

Step-by-step solution

Part a· 4 steps

Substitute $x=4$ into the function $h(x) = bx - 5$ .

Use the given information that $h(4) = -1$ to set up an equation.

Add 5 to both sides of the equation.

Divide by 4 to solve for $b$ .

Part b· 3 stepsx+1

Factor the numerator $x^2 - 1$ using the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ .

Substitute the factored numerator back into the expression for $g(x)$ .

Cancel out the common term $(x-1)$ from the numerator and the denominator.

Part c· 2 steps
$\text{all real numbers except } x=1$

Identify any values of $x$ that would make the denominator of the original function $g(x) = \frac{x^2 - 1}{x - 1}$ equal to zero.

The domain of definition includes all real numbers except for the values that make the denominator zero.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Solve the following questions: a) If h(x) = bx − 5 and h(4) = -1, find b. b) Simplify g(x) = (x2 − 1)/(x - 1). c) State the domain of definition of g(x).

Answers

Step-by-step solution

Part a· 4 steps

Substitute x=4x=4x=4 into the function h(x)=bx−5h(x) = bx - 5h(x)=bx−5.

Use the given information that h(4)=−1h(4) = -1h(4)=−1 to set up an equation.