The functions f and g are defined as f:x 2192 x − 2 and g:x 2192 2x2 - 1. Solve: i. f(x) = g(−1/2) ii. f(x)…

Question

The functions f and g are defined as f:x 2192 x − 2 and g:x 2192 2x2 - 1. Solve: i. f(x) = g(−1/2) ii. f(x) + g(x) = 0

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for 5a.i and 5a.ii.

5a.i. Solve $f(x) = g\left(-\frac{1}{2}\right)$

Identify the functions $f(x)$ and $g(x)$ .
We are given $f(x) = x-2$ and $g(x) = 2x^2 - 1$ .
Evaluate $g\left(-\frac{1}{2}\right)$ .
Substitute $x = -\frac{1}{2}$ into the expression for $g(x)$ :
$g\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right)^2 - 1$ $g\left(-\frac{1}{2}\right) = 2\left(\frac{1}{4}\right) - 1$ $g\left(-\frac{1}{2}\right) = \frac{1}{2} - 1$ $g\left(-\frac{1}{2}\right) = -\frac{1}{2}$
Set $f(x)$ equal to the value found for $g\left(-\frac{1}{2}\right)$ .
$f(x) = -\frac{1}{2}$ $x-2 = -\frac{1}{2}$
Solve for $x$ .
Add 2 to both sides of the equation:
$x = 2 - \frac{1}{2}$ $x = \frac{4}{2} - \frac{1}{2}$ $x = \frac{3}{2}$
The solution is $\boxed{x = \frac{3}{2}}$ .

5a.ii. Solve $f(x) + g(x) = 0$
Substitute the expressions for $f(x)$ and $g(x)$ into the equation.
$(x-2) + (2x^2 - 1) = 0$
Simplify and rearrange the equation into standard quadratic form $ax^2 + bx + c = 0$ .
$2x^2 + x - 2 - 1 = 0$ $2x^2 + x - 3 = 0$
Factor the quadratic equation.
We look for two numbers that multiply to $2 \times (-3) = -6$ and add to 1. These numbers are 3 and $-2$ .
$2x^2 + 3x - 2x - 3 = 0$
Factor by grouping:
$x(2x+3) - 1(2x+3) = 0$ $(x-1)(2x+3) = 0$
Set each factor to zero and solve for $x$ .
$x-1 = 0 \implies x = 1$ $2x+3 = 0 \implies 2x = -3 \implies x = -\frac{3}{2}$
The solutions are $\boxed{x = 1 \text{ or } x = -\frac{3}{2}}$ .

The functions f and g are defined as f:x 2192 x - 2 and g:x 2192 2x2 - 1. Solve: i. f(x) = g(-1/2) ii. f(x) + g(x) = 0