Question

Find all vertical asymptotes of the following function. f(x)=3ˣ²-14x+16)/(2ˣ²-10x+12)

Answer 1

The vertical asymptotes of the function
`\( f(x) = (3^(x^2) - 14x + 16)/(2^(x^2) - 10x + 12) \) are \( x = 2 \) and \( x = 3 \).`

Step-by-step explanation

Vertical asymptotes occur when the denominator of a rational function equals zero, leading to undefined values. To find these points for f(x) , set the denominator
`\( 2^(x^2) - 10x + 12 \)` equal to zero and solve for x . Factoring the quadratic expression yields
`\( (2^x - 6)(2^x - 2) = 0 \)`. Setting each factor equal to zero gives
`\( 2^x - 6 = 0 \) and \( 2^x - 2 = 0 \).`Solving these equations, we find
`\( x = 3 \) and \( x = 2 \) a`s the values at which the denominator becomes zero.

Now, let's consider the explanation for these results. When \x = 3 , the denominator becomes
`\( 2^9 - 10 * 3 + 12 = 0 \)`. Similarly, when x = 2 , the denominator becomes
`\( 2^4 - 10 * 2 + 12 = 0 \).` In both cases, the denominator equals zero, indicating vertical asymptotes at
`\( x = 3 \) and \( x = 2 \). As \( x \)`approaches these values, the function f(x) tends toward positive or negative infinity, signifying the presence of vertical asymptotes at these points. Thus, the identified vertical asymptotes for the given function are
`\( x = 2 \) and \( x = 3 \).`