Math Snap

STEP 1

1. We are given two functions: $g(x) = 4x - 5$ and $t(x) = x^2 - 5x + 1$.
2. We need to evaluate $t(6)$.
3. We need to show that $t(g(x)) = 16x^2 - 60x + 51$.

STEP 2

1. Evaluate $t(6)$.
2. Substitute $g(x)$ into $t(x)$ to find $t(g(x))$.
3. Simplify the expression for $t(g(x))$.
4. Verify that the simplified expression matches $16x^2 - 60x + 51$.

STEP 3

Substitute $x = 6$ into $t(x)$:
$$t(6) = 6^2 - 5(6) + 1$$

STEP 4

Simplify the expression:
$$t(6) = 36 - 30 + 1$$ $$t(6) = 7$$

STEP 5

Substitute $g(x) = 4x - 5$ into $t(x)$:
$$t(g(x)) = ((4x - 5)^2) - 5(4x - 5) + 1$$

STEP 6

Expand $(4x - 5)^2$:
$$(4x - 5)^2 = (4x)^2 - 2(4x)(5) + 5^2$$ $$= 16x^2 - 40x + 25$$

STEP 7

Substitute the expanded form back into the expression for $t(g(x))$:
$$t(g(x)) = (16x^2 - 40x + 25) - 5(4x - 5) + 1$$

STEP 8

Simplify the expression:
$$t(g(x)) = 16x^2 - 40x + 25 - (20x - 25) + 1$$

STEP 9

Combine like terms:
$$t(g(x)) = 16x^2 - 40x + 25 - 20x + 25 + 1$$ $$t(g(x)) = 16x^2 - 60x + 51$$

Verify that the expression matches $16x^2 - 60x + 51$.
The expression matches, confirming the result.
The value of $t(6)$ is $\boxed{7}$, and it is shown that $t(g(x)) = 16x^2 - 60x + 51$.